(sort "wff") ;; (_Note_: This inference rule and the next one, ~ idi , will normally never appear in a completed proof. They can be ignored if you are using this database to assist learning logic - please start with the statement ~ wn instead.) This is a technical inference to assist proof development. It provides a temporary way to add an independent subproof to a proof under development, for later assignment to a normal proof step. The metamath program's Proof Assistant requires proofs to be developed backwards from the conclusion with no gaps, and it has no mechanism that lets the user to work on isolated subproofs. This inference provides a workaround for this limitation. It can be inserted at any point in a proof to allow an independent subproof to be developed on the side, for later use as part of the final proof. _Instructions_: (1) Assign this inference to any unknown step in the proof. Typically, the last unknown step is the most convenient, since 'assign last' can be used. This step will be replicated in hypothesis a1ii.1, from where the development of the main proof can continue. (2) Develop the independent subproof backwards from hypothesis a1ii.2. If desired, use a 'let' command to pre-assign the conclusion of the independent subproof to a1ii.2. (3) After the independent subproof is complete, use 'improve all' to assign it automatically to an unknown step in the main proof that matches it. (4) After the entire proof is complete, use 'minimize *' to clean up (discard) all ~ a1ii references automatically. This inference was originally designed to assist importing partially completed Proof Worksheets from the mmj2 Proof Assistant GUI, but it can also be useful on its own. Interestingly, no axioms are required for its proof. It is the inference associated with ~ a1i . (Contributed by NM, 7-Feb-2006.) (theorem "a1ii" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("a1ii_1" "ph") ("a1ii_2" "ps")) (for) "ph" "a1ii_1") ;; This inference, which requires no axioms for its proof, is useful as a copy-paste mechanism during proof development in mmj2. It is normally not referenced in the final version of a proof, since it is always redundant and can be removed using the 'minimize *' command in the metamath program's Proof Assistant. It is the inference associated with ~ id . (Contributed by Alan Sare, 31-Dec-2011.) (theorem "idi" (for ("ph" ( "wff"))) (for ("idi_1" "ph")) (for) "ph" "idi_1") ;; If ` ph ` is a wff, so is ` -. ph ` or "not ` ph ` ." Part of the recursive definition of a wff (well-formed formula). In classical logic (which is our logic), a wff is interpreted as either true or false. So if ` ph ` is true, then ` -. ph ` is false; if ` ph ` is false, then ` -. ph ` is true. Traditionally, Greek letters are used to represent wffs, and we follow this convention. In propositional calculus, we define only wffs built up from other wffs, i.e. there is no starting or "atomic" wff. Later, in predicate calculus, we will extend the basic wff definition by including atomic wffs ( ~ weq and ~ wel ). (term "wn" ( ( "wff") ( "wff"))) ;; If ` ph ` and ` ps ` are wff's, so is ` ( ph -> ps ) ` or " ` ph ` implies ` ps ` ." Part of the recursive definition of a wff. The resulting wff is (interpreted as) false when ` ph ` is true and ` ps ` is false; it is true otherwise. Think of the truth table for an OR gate with input ` ph ` connected through an inverter. After we define the axioms of propositional calculus ( ~ ax-1 , ~ ax-2 , ~ ax-3 , and ~ ax-mp ), the biconditional ( ~ df-bi ), the constant true ` T. ` ( ~ df-tru ), and the constant false ` F. ` ( ~ df-fal ), we will be able to prove these truth table values: ` ( ( T. -> T. ) <-> T. ) ` ( ~ truimtru ), ` ( ( T. -> F. ) <-> F. ) ` ( ~ truimfal ), ` ( ( F. -> T. ) <-> T. ) ` ( ~ falimtru ), and ` ( ( F. -> F. ) <-> T. ) ` ( ~ falimfal ). These have straightforward meanings, for example, ` ( ( T. -> T. ) <-> T. ) ` just means "the value of ` ( T. -> T. ) ` is ` T. ` ". The left-hand wff is called the antecedent, and the right-hand wff is called the consequent. In the case of ` ( ph -> ( ps -> ch ) ) ` , the middle ` ps ` may be informally called either an antecedent or part of the consequent depending on context. Contrast with ` <-> ` ( ~ df-bi ), ` /\ ` ( ~ df-an ), and ` \/ ` ( ~ df-or ). This is called "material implication" and the arrow is usually read as "implies." However, material implication is not identical to the meaning of "implies" in natural language. For example, the word "implies" may suggest a causal relationship in natural language. Material implication does not require any causal relationship. Also, note that in material implication, if the consequent is true then the wff is always true (even if the antecedent is false). Thus, if "implies" means material implication, it is true that "if the moon is made of green cheese that implies that 5=5" (because 5=5). Similarly, if the antecedent is false, the wff is always true. Thus, it is true that, "if the moon is made of green cheese that implies that 5=7" (because the moon is not actually made of green cheese). A contradiction implies anything ( ~ pm2.21i ). In short, material implication has a very specific technical definition, and misunderstandings of it are sometimes called "paradoxes of logical implication." (term "wi" ( ( "wff") ( "wff") ( "wff"))) ;; Rule of Modus Ponens. The postulated inference rule of propositional calculus. See e.g. Rule 1 of [Hamilton] p. 73. The rule says, "if ` ph ` is true, and ` ph ` implies ` ps ` , then ` ps ` must also be true." This rule is sometimes called "detachment," since it detaches the minor premise from the major premise. "Modus ponens" is short for "modus ponendo ponens," a Latin phrase that means "the mode that by affirming affirms" - remark in [Sanford] p. 39. This rule is similar to the rule of modus tollens ~ mto . Note: In some web page displays such as the Statement List, the symbols " ` & ` " and " ` => ` " informally indicate the relationship between the hypotheses and the assertion (conclusion), abbreviating the English words "and" and "implies." They are not part of the formal language. (Contributed by NM, 30-Sep-1992.) (axiom "ax_mp" (!! ( ("ph" ( "wff")) ("ps" ( "wff"))) "ph" ("wi" "ph" "ps") "ps")) ;; Axiom _Simp_. Axiom A1 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. The 3 axioms are also given as Definition 2.1 of [Hamilton] p. 28. This axiom is called _Simp_ or "the principle of simplification" in _Principia Mathematica_ (Theorem *2.02 of [WhiteheadRussell] p. 100) because "it enables us to pass from the joint assertion of ` ph ` and ` ps ` to the assertion of ` ph ` simply." It is Proposition 1 of [Frege1879] p. 26, its first axiom. (Contributed by NM, 30-Sep-1992.) (axiom "ax_1" (!! ( ("ph" ( "wff")) ("ps" ( "wff"))) ("wi" "ph" ("wi" "ps" "ph")))) ;; Axiom _Frege_. Axiom A2 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It "distributes" an antecedent over two consequents. This axiom was part of Frege's original system and is known as _Frege_ in the literature; see Proposition 2 of [Frege1879] p. 26. It is also proved as Theorem *2.77 of [WhiteheadRussell] p. 108. The other direction of this axiom also turns out to be true, as demonstrated by ~ pm5.41 . (Contributed by NM, 30-Sep-1992.) (axiom "ax_2" (!! ( ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) ("wi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch"))))) ;; Axiom _Transp_. Axiom A3 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky." This axiom is called _Transp_ or "the principle of transposition" in _Principia Mathematica_ (Theorem *2.17 of [WhiteheadRussell] p. 103). We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning. (Contributed by NM, 30-Sep-1992.) Use its alias ~ con4 instead. (New usage is discouraged.) (axiom "ax_3" (!! ( ("ph" ( "wff")) ("ps" ( "wff"))) ("wi" ("wi" ("wn" "ph") ("wn" "ps")) ("wi" "ps" "ph")))) ;; A double modus ponens inference. (Contributed by NM, 5-Apr-1994.) (theorem "mp2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mp2_1" "ph") ("mp2_2" "ps") ("mp2_3" ("wi" "ph" ("wi" "ps" "ch")))) (for) "ch" ("ax_mp" "ps" "ch" "mp2_2" ("ax_mp" "ph" ("wi" "ps" "ch") "mp2_1" "mp2_3"))) ;; A double modus ponens inference. (Contributed by Mario Carneiro, 24-Jan-2013.) (theorem "mp2b" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mp2b_1" "ph") ("mp2b_2" ("wi" "ph" "ps")) ("mp2b_3" ("wi" "ps" "ch"))) (for) "ch" ("ax_mp" "ps" "ch" ("ax_mp" "ph" "ps" "mp2b_1" "mp2b_2") "mp2b_3")) ;; Inference introducing an antecedent. Inference associated with ~ ax-1 . Its associated inference is ~ a1ii . See ~ conventions for a definition of "associated inference". (Contributed by NM, 29-Dec-1992.) (theorem "a1i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("a1i_1" "ph")) (for) ("wi" "ps" "ph") ("ax_mp" "ph" ("wi" "ps" "ph") "a1i_1" ("ax_1" "ph" "ps"))) ;; Inference introducing two antecedents. Two applications of ~ a1i . Inference associated with ~ 2a1 . (Contributed by Jeff Hankins, 4-Aug-2009.) (theorem "_2a1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_2a1i_1" "ph")) (for) ("wi" "ps" ("wi" "ch" "ph")) ("a1i" ("wi" "ch" "ph") "ps" ("a1i" "ph" "ch" "_2a1i_1"))) ;; Inference detaching an antecedent and introducing a new one. (Contributed by Stefan O'Rear, 29-Jan-2015.) (theorem "mp1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mp1i_1" "ph") ("mp1i_2" ("wi" "ph" "ps"))) (for) ("wi" "ch" "ps") ("a1i" "ps" "ch" ("ax_mp" "ph" "ps" "mp1i_1" "mp1i_2"))) ;; Inference distributing an antecedent. Inference associated with ~ ax-2 . Its associated inference is ~ mpd . (Contributed by NM, 29-Dec-1992.) (theorem "a2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("a2i_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("ax_mp" ("wi" "ph" ("wi" "ps" "ch")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch")) "a2i_1" ("ax_2" "ph" "ps" "ch"))) ;; A modus ponens deduction. A translation of natural deduction rule ` -> ` E ( ` -> ` elimination), see ~ natded . Deduction form of ~ ax-mp . Inference associated with ~ a2i . Commuted form of ~ mpcom . (Contributed by NM, 29-Dec-1992.) (theorem "mpd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpd_1" ("wi" "ph" "ps")) ("mpd_2" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" "ch") ("ax_mp" ("wi" "ph" "ps") ("wi" "ph" "ch") "mpd_1" ("a2i" "ph" "ps" "ch" "mpd_2"))) ;; Inference adding common antecedents in an implication. Inference associated with ~ imim2 . Its associated inference is ~ syl . (Contributed by NM, 28-Dec-1992.) (theorem "imim2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("imim2i_1" ("wi" "ph" "ps"))) (for) ("wi" ("wi" "ch" "ph") ("wi" "ch" "ps")) ("a2i" "ch" "ph" "ps" ("a1i" ("wi" "ph" "ps") "ch" "imim2i_1"))) ;; An inference version of the transitive laws for implication ~ imim2 and ~ imim1 (and ~ imim1i and ~ imim2i ), which Russell and Whitehead call "the principle of the syllogism...because...the syllogism in Barbara is derived from them" (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Some authors call this law a "hypothetical syllogism." Its associated inference is ~ mp2b . (A bit of trivia: this is the most commonly referenced assertion in our database (13449 times as of 22-Jul-2021). In second place is ~ eqid (9597 times), followed by ~ adantr (8861 times), ~ syl2anc (7421 times), ~ adantl (6403 times), and ~ simpr (5829 times). The Metamath program command 'show usage' shows the number of references.) (Contributed by NM, 30-Sep-1992.) (Proof shortened by Mel L. O'Cat, 20-Oct-2011.) (Proof shortened by Wolf Lammen, 26-Jul-2012.) (theorem "syl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("syl_1" ("wi" "ph" "ps")) ("syl_2" ("wi" "ps" "ch"))) (for) ("wi" "ph" "ch") ("mpd" "ph" "ps" "ch" "syl_1" ("a1i" ("wi" "ps" "ch") "ph" "syl_2"))) ;; Inference chaining two syllogisms ~ syl . Inference associated with ~ imim12i . (Contributed by NM, 28-Dec-1992.) (theorem "_3syl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3syl_1" ("wi" "ph" "ps")) ("_3syl_2" ("wi" "ps" "ch")) ("_3syl_3" ("wi" "ch" "th"))) (for) ("wi" "ph" "th") ("syl" "ph" "ch" "th" ("syl" "ph" "ps" "ch" "_3syl_1" "_3syl_2") "_3syl_3")) ;; Inference chaining three syllogisms ~ syl . (Contributed by BJ, 14-Jul-2018.) The use of this theorem is marked "discouraged" because it can cause the "minimize" command to have very long run times. However, feel free to use "minimize 4syl /override" if you wish. Remember to update the Travis "discouraged" file if it gets used. (New usage is discouraged.) (theorem "_4syl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_4syl_1" ("wi" "ph" "ps")) ("_4syl_2" ("wi" "ps" "ch")) ("_4syl_3" ("wi" "ch" "th")) ("_4syl_4" ("wi" "th" "ta"))) (for) ("wi" "ph" "ta") ("syl" "ph" "th" "ta" ("_3syl" "ph" "ps" "ch" "th" "_4syl_1" "_4syl_2" "_4syl_3") "_4syl_4")) ;; A nested modus ponens inference. Inference associated with ~ com12 . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Stefan Allan, 20-Mar-2006.) (theorem "mpi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpi_1" "ps") ("mpi_2" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" "ch") ("mpd" "ph" "ps" "ch" ("a1i" "ps" "ph" "mpi_1") "mpi_2")) ;; A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.) (theorem "mpisyl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpisyl_1" ("wi" "ph" "ps")) ("mpisyl_2" "ch") ("mpisyl_3" ("wi" "ps" ("wi" "ch" "th")))) (for) ("wi" "ph" "th") ("syl" "ph" "ps" "th" "mpisyl_1" ("mpi" "ps" "ch" "th" "mpisyl_2" "mpisyl_3"))) ;; Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. For another version of the proof directly from axioms, see ~ idALT . Its associated inference, ~ idi , requires no axioms for its proof, contrary to ~ id . Note that the second occurrences of ` ph ` in Steps 1 and 2 may be simultaneously replaced by any wff ` ps ` , which may ease the understanding of the proof. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Stefan Allan, 20-Mar-2006.) (theorem "id" (for ("ph" ( "wff"))) (for) (for) ("wi" "ph" "ph") ("mpd" "ph" ("wi" "ph" "ph") "ph" ("ax_1" "ph" "ph") ("ax_1" "ph" ("wi" "ph" "ph")))) ;; Alternate proof of ~ id . This version is proved directly from the axioms for demonstration purposes. This proof is a popular example in the literature and is identical, step for step, to the proofs of Theorem 1 of [Margaris] p. 51, Example 2.7(a) of [Hamilton] p. 31, Lemma 10.3 of [BellMachover] p. 36, and Lemma 1.8 of [Mendelson] p. 36. It is also "Our first proof" in Hirst and Hirst's _A Primer for Logic and Proof_ p. 17 (PDF p. 23) at ~ http://www.appstate.edu/~~hirstjl/primer/hirst.pdf . Note that the second occurrences of ` ph ` in Steps 1 to 4 and the sixth in Step 3 may be simultaneously replaced by any wff ` ps ` , which may ease the understanding of the proof. For a shorter version of the proof that takes advantage of previously proved theorems, see ~ id . (Contributed by NM, 30-Sep-1992.) (Proof modification is discouraged.) Use ~ id instead. (New usage is discouraged.) (theorem "idALT" (for ("ph" ( "wff"))) (for) (for) ("wi" "ph" "ph") ("ax_mp" ("wi" "ph" ("wi" "ph" "ph")) ("wi" "ph" "ph") ("ax_1" "ph" "ph") ("ax_mp" ("wi" "ph" ("wi" ("wi" "ph" "ph") "ph")) ("wi" ("wi" "ph" ("wi" "ph" "ph")) ("wi" "ph" "ph")) ("ax_1" "ph" ("wi" "ph" "ph")) ("ax_2" "ph" ("wi" "ph" "ph") "ph")))) ;; Principle of identity ~ id with antecedent. (Contributed by NM, 26-Nov-1995.) (theorem "idd" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wi" "ps" "ps")) ("a1i" ("wi" "ps" "ps") "ph" ("id" "ps"))) ;; Deduction introducing an embedded antecedent. Deduction form of ~ ax-1 and ~ a1i . (Contributed by NM, 5-Jan-1993.) (Proof shortened by Stefan Allan, 20-Mar-2006.) (theorem "a1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("a1d_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("wi" "ch" "ps")) ("syl" "ph" "ps" ("wi" "ch" "ps") "a1d_1" ("ax_1" "ps" "ch"))) ;; Deduction introducing two antecedents. Two applications of ~ a1d . Deduction associated with ~ 2a1 and ~ 2a1i . (Contributed by BJ, 10-Aug-2020.) (theorem "_2a1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_2a1d_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("wi" "ch" ("wi" "th" "ps"))) ("a1d" "ph" ("wi" "th" "ps") "ch" ("a1d" "ph" "ps" "th" "_2a1d_1"))) ;; Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.) (theorem "a1i13" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("a1i13_1" ("wi" "ps" "th"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))) ("a1i" ("wi" "ps" ("wi" "ch" "th")) "ph" ("a1d" "ps" "th" "ch" "a1i13_1"))) ;; A double form of ~ ax-1 . Its associated inference is ~ 2a1i . Its associated deduction is ~ 2a1d . (Contributed by BJ, 10-Aug-2020.) (Proof shortened by Wolf Lammen, 1-Sep-2020.) (theorem "_2a1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "ph"))) ("_2a1d" "ph" "ph" "ps" "ch" ("id" "ph"))) ;; Deduction distributing an embedded antecedent. Deduction form of ~ ax-2 . (Contributed by NM, 23-Jun-1994.) (theorem "a2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("a2d_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ph" ("wi" ("wi" "ps" "ch") ("wi" "ps" "th"))) ("syl" "ph" ("wi" "ps" ("wi" "ch" "th")) ("wi" ("wi" "ps" "ch") ("wi" "ps" "th")) "a2d_1" ("ax_2" "ps" "ch" "th"))) ;; Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.) (theorem "sylcom" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylcom_1" ("wi" "ph" ("wi" "ps" "ch"))) ("sylcom_2" ("wi" "ps" ("wi" "ch" "th")))) (for) ("wi" "ph" ("wi" "ps" "th")) ("syl" "ph" ("wi" "ps" "ch") ("wi" "ps" "th") "sylcom_1" ("a2i" "ps" "ch" "th" "sylcom_2"))) ;; Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.) (theorem "syl5com" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5com_1" ("wi" "ph" "ps")) ("syl5com_2" ("wi" "ch" ("wi" "ps" "th")))) (for) ("wi" "ph" ("wi" "ch" "th")) ("sylcom" "ph" "ch" "ps" "th" ("a1d" "ph" "ps" "ch" "syl5com_1") "syl5com_2")) ;; Inference that swaps (commutes) antecedents in an implication. Inference associated with ~ pm2.04 . Its associated inference is ~ mpi . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.) (theorem "com12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("com12_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ps" ("wi" "ph" "ch")) ("syl5com" "ps" "ps" "ph" "ch" ("id" "ps") "com12_1")) ;; A syllogism inference. Commuted form of an instance of ~ syl . (Contributed by BJ, 25-Oct-2021.) (theorem "syl11" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl11_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl11_2" ("wi" "th" "ph"))) (for) ("wi" "ps" ("wi" "th" "ch")) ("com12" "th" "ps" "ch" ("syl" "th" "ph" ("wi" "ps" "ch") "syl11_2" "syl11_1"))) ;; A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.) (theorem "syl5" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5_1" ("wi" "ph" "ps")) ("syl5_2" ("wi" "ch" ("wi" "ps" "th")))) (for) ("wi" "ch" ("wi" "ph" "th")) ("com12" "ph" "ch" "th" ("syl5com" "ph" "ps" "ch" "th" "syl5_1" "syl5_2"))) ;; A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Wolf Lammen, 30-Jul-2012.) (theorem "syl6" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl6_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl6_2" ("wi" "ch" "th"))) (for) ("wi" "ph" ("wi" "ps" "th")) ("sylcom" "ph" "ps" "ch" "th" "syl6_1" ("a1i" ("wi" "ch" "th") "ps" "syl6_2"))) ;; Combine ~ syl5 and ~ syl6 . (Contributed by NM, 14-Nov-2013.) (theorem "syl56" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl56_1" ("wi" "ph" "ps")) ("syl56_2" ("wi" "ch" ("wi" "ps" "th"))) ("syl56_3" ("wi" "th" "ta"))) (for) ("wi" "ch" ("wi" "ph" "ta")) ("syl5" "ph" "ps" "ch" "ta" "syl56_1" ("syl6" "ch" "ps" "th" "ta" "syl56_2" "syl56_3"))) ;; Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.) (theorem "syl6com" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl6com_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl6com_2" ("wi" "ch" "th"))) (for) ("wi" "ps" ("wi" "ph" "th")) ("com12" "ph" "ps" "th" ("syl6" "ph" "ps" "ch" "th" "syl6com_1" "syl6com_2"))) ;; Modus ponens inference with commutation of antecedents. Commuted form of ~ mpd . (Contributed by NM, 17-Mar-1996.) (theorem "mpcom" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpcom_1" ("wi" "ps" "ph")) ("mpcom_2" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ps" "ch") ("mpd" "ps" "ph" "ch" "mpcom_1" ("com12" "ph" "ps" "ch" "mpcom_2"))) ;; Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.) (theorem "syli" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syli_1" ("wi" "ps" ("wi" "ph" "ch"))) ("syli_2" ("wi" "ch" ("wi" "ph" "th")))) (for) ("wi" "ps" ("wi" "ph" "th")) ("sylcom" "ps" "ph" "ch" "th" "syli_1" ("com12" "ch" "ph" "th" "syli_2"))) ;; Replace two antecedents. Implication-only version of ~ syl2an . (Contributed by Wolf Lammen, 14-May-2013.) (theorem "syl2im" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl2im_1" ("wi" "ph" "ps")) ("syl2im_2" ("wi" "ch" "th")) ("syl2im_3" ("wi" "ps" ("wi" "th" "ta")))) (for) ("wi" "ph" ("wi" "ch" "ta")) ("syl" "ph" "ps" ("wi" "ch" "ta") "syl2im_1" ("syl5" "ch" "th" "ps" "ta" "syl2im_2" "syl2im_3"))) ;; A commuted version of ~ syl2im . Implication-only version of ~ syl2anr . (Contributed by BJ, 20-Oct-2021.) (theorem "syl2imc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl2im_1" ("wi" "ph" "ps")) ("syl2im_2" ("wi" "ch" "th")) ("syl2im_3" ("wi" "ps" ("wi" "th" "ta")))) (for) ("wi" "ch" ("wi" "ph" "ta")) ("com12" "ph" "ch" "ta" ("syl2im" "ph" "ps" "ch" "th" "ta" "syl2im_1" "syl2im_2" "syl2im_3"))) ;; This theorem, called "Assertion," can be thought of as closed form of modus ponens ~ ax-mp . Theorem *2.27 of [WhiteheadRussell] p. 104. (Contributed by NM, 15-Jul-1993.) (theorem "pm2_27" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wi" ("wi" "ph" "ps") "ps")) ("com12" ("wi" "ph" "ps") "ph" "ps" ("id" ("wi" "ph" "ps")))) ;; A nested modus ponens deduction. Double deduction associated with ~ ax-mp . Deduction associated with ~ mpd . (Contributed by NM, 12-Dec-2004.) (theorem "mpdd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpdd_1" ("wi" "ph" ("wi" "ps" "ch"))) ("mpdd_2" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ph" ("wi" "ps" "th")) ("mpd" "ph" ("wi" "ps" "ch") ("wi" "ps" "th") "mpdd_1" ("a2d" "ph" "ps" "ch" "th" "mpdd_2"))) ;; A nested modus ponens deduction. Deduction associated with ~ mpi . (Contributed by NM, 14-Dec-2004.) (theorem "mpid" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpid_1" ("wi" "ph" "ch")) ("mpid_2" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ph" ("wi" "ps" "th")) ("mpdd" "ph" "ps" "ch" "th" ("a1d" "ph" "ch" "ps" "mpid_1") "mpid_2")) ;; A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.) (theorem "mpdi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpdi_1" ("wi" "ps" "ch")) ("mpdi_2" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ph" ("wi" "ps" "th")) ("mpdd" "ph" "ps" "ch" "th" ("a1i" ("wi" "ps" "ch") "ph" "mpdi_1") "mpdi_2")) ;; A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.) (theorem "mpii" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpii_1" "ch") ("mpii_2" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ph" ("wi" "ps" "th")) ("mpdi" "ph" "ps" "ch" "th" ("a1i" "ch" "ps" "mpii_1") "mpii_2")) ;; Syllogism deduction. Deduction associated with ~ syl . See ~ conventions for the meaning of "associated deduction" or "deduction form". (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.) (theorem "syld" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syld_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syld_2" ("wi" "ph" ("wi" "ch" "th")))) (for) ("wi" "ph" ("wi" "ps" "th")) ("mpdd" "ph" "ps" "ch" "th" "syld_1" ("a1d" "ph" ("wi" "ch" "th") "ps" "syld_2"))) ;; Syllogism deduction. Commuted form of ~ syld . (Contributed by BJ, 25-Oct-2021.) (theorem "syldc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syld_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syld_2" ("wi" "ph" ("wi" "ch" "th")))) (for) ("wi" "ps" ("wi" "ph" "th")) ("com12" "ph" "ps" "th" ("syld" "ph" "ps" "ch" "th" "syld_1" "syld_2"))) ;; A double modus ponens deduction. Deduction associated with ~ mp2 . (Contributed by NM, 23-May-2013.) (Proof shortened by Wolf Lammen, 23-Jul-2013.) (theorem "mp2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mp2d_1" ("wi" "ph" "ps")) ("mp2d_2" ("wi" "ph" "ch")) ("mp2d_3" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ph" "th") ("mpd" "ph" "ps" "th" "mp2d_1" ("mpid" "ph" "ps" "ch" "th" "mp2d_2" "mp2d_3"))) ;; Double deduction introducing an antecedent. Deduction associated with ~ a1d . Double deduction associated with ~ ax-1 and ~ a1i . (Contributed by NM, 17-Dec-2004.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.) (theorem "a1dd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("a1dd_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "th" "ch"))) ("syl6" "ph" "ps" "ch" ("wi" "th" "ch") "a1dd_1" ("ax_1" "ch" "th"))) ;; Double deduction introducing two antecedents. Two applications of ~ 2a1dd . Deduction associated with ~ 2a1d . Double deduction associated with ~ 2a1 and ~ 2a1i . (Contributed by Jeff Hankins, 5-Aug-2009.) (theorem "_2a1dd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_2a1dd_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "th" ("wi" "ta" "ch")))) ("a1dd" "ph" "ps" ("wi" "ta" "ch") "th" ("a1dd" "ph" "ps" "ch" "ta" "_2a1dd_1"))) ;; Inference absorbing redundant antecedent. Inference associated with ~ pm2.43 . (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (theorem "pm2_43i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm2_43i_1" ("wi" "ph" ("wi" "ph" "ps")))) (for) ("wi" "ph" "ps") ("mpd" "ph" "ph" "ps" ("id" "ph") "pm2_43i_1")) ;; Deduction absorbing redundant antecedent. Deduction associated with ~ pm2.43 and ~ pm2.43i . (Contributed by NM, 18-Aug-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (theorem "pm2_43d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_43d_1" ("wi" "ph" ("wi" "ps" ("wi" "ps" "ch"))))) (for) ("wi" "ph" ("wi" "ps" "ch")) ("mpdi" "ph" "ps" "ps" "ch" ("id" "ps") "pm2_43d_1")) ;; Inference absorbing redundant antecedent. (Contributed by NM, 7-Nov-1995.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (theorem "pm2_43a" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_43a_1" ("wi" "ps" ("wi" "ph" ("wi" "ps" "ch"))))) (for) ("wi" "ps" ("wi" "ph" "ch")) ("mpid" "ps" "ph" "ps" "ch" ("id" "ps") "pm2_43a_1")) ;; Inference absorbing redundant antecedent. (Contributed by NM, 31-Oct-1995.) (theorem "pm2_43b" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_43b_1" ("wi" "ps" ("wi" "ph" ("wi" "ps" "ch"))))) (for) ("wi" "ph" ("wi" "ps" "ch")) ("com12" "ps" "ph" "ch" ("pm2_43a" "ph" "ps" "ch" "pm2_43b_1"))) ;; Absorption of redundant antecedent. Also called the "Contraction" or "Hilbert" axiom. Theorem *2.43 of [WhiteheadRussell] p. 106. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 15-Aug-2004.) (theorem "pm2_43" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" "ph" ("wi" "ph" "ps")) ("wi" "ph" "ps")) ("a2i" "ph" ("wi" "ph" "ps") "ps" ("pm2_27" "ph" "ps"))) ;; Deduction adding nested antecedents. Deduction associated with ~ imim2 and ~ imim2i . (Contributed by NM, 10-Jan-1993.) (theorem "imim2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("imim2d_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" ("wi" "th" "ps") ("wi" "th" "ch"))) ("a2d" "ph" "th" "ps" "ch" ("a1d" "ph" ("wi" "ps" "ch") "th" "imim2d_1"))) ;; A closed form of syllogism (see ~ syl ). Theorem *2.05 of [WhiteheadRussell] p. 100. Its associated inference is ~ imim2i . Its associated deduction is ~ imim2d . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 6-Sep-2012.) (theorem "imim2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wi" "ch" "ph") ("wi" "ch" "ps"))) ("imim2d" ("wi" "ph" "ps") "ph" "ps" "ch" ("id" ("wi" "ph" "ps")))) ;; Deduction embedding an antecedent. (Contributed by Wolf Lammen, 4-Oct-2013.) (theorem "embantd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("embantd_1" ("wi" "ph" "ps")) ("embantd_2" ("wi" "ph" ("wi" "ch" "th")))) (for) ("wi" "ph" ("wi" ("wi" "ps" "ch") "th")) ("mpid" "ph" ("wi" "ps" "ch") "ps" "th" "embantd_1" ("imim2d" "ph" "ch" "th" "ps" "embantd_2"))) ;; Triple syllogism deduction. Deduction associated with ~ 3syld . (Contributed by Jeff Hankins, 4-Aug-2009.) (theorem "_3syld" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3syld_1" ("wi" "ph" ("wi" "ps" "ch"))) ("_3syld_2" ("wi" "ph" ("wi" "ch" "th"))) ("_3syld_3" ("wi" "ph" ("wi" "th" "ta")))) (for) ("wi" "ph" ("wi" "ps" "ta")) ("syld" "ph" "ps" "th" "ta" ("syld" "ph" "ps" "ch" "th" "_3syld_1" "_3syld_2") "_3syld_3")) ;; A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.) (theorem "sylsyld" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylsyld_1" ("wi" "ph" "ps")) ("sylsyld_2" ("wi" "ph" ("wi" "ch" "th"))) ("sylsyld_3" ("wi" "ps" ("wi" "th" "ta")))) (for) ("wi" "ph" ("wi" "ch" "ta")) ("syld" "ph" "ch" "th" "ta" "sylsyld_2" ("syl" "ph" "ps" ("wi" "th" "ta") "sylsyld_1" "sylsyld_3"))) ;; Inference joining two implications. Inference associated with ~ imim12 . Its associated inference is ~ 3syl . (Contributed by NM, 12-Mar-1993.) (Proof shortened by Mel L. O'Cat, 29-Oct-2011.) (theorem "imim12i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("imim12i_1" ("wi" "ph" "ps")) ("imim12i_2" ("wi" "ch" "th"))) (for) ("wi" ("wi" "ps" "ch") ("wi" "ph" "th")) ("syl5" "ph" "ps" ("wi" "ps" "ch") "th" "imim12i_1" ("imim2i" "ch" "th" "ps" "imim12i_2"))) ;; Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. Inference associated with ~ imim1 . Its associated inference is ~ syl . (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.) (theorem "imim1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("imim1i_1" ("wi" "ph" "ps"))) (for) ("wi" ("wi" "ps" "ch") ("wi" "ph" "ch")) ("imim12i" "ph" "ps" "ch" "ch" "imim1i_1" ("id" "ch"))) ;; Inference adding three nested antecedents. (Contributed by NM, 19-Dec-2006.) (theorem "imim3i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("imim3i_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" ("wi" "th" "ph") ("wi" ("wi" "th" "ps") ("wi" "th" "ch"))) ("a2d" ("wi" "th" "ph") "th" "ps" "ch" ("imim2i" "ph" ("wi" "ps" "ch") "th" "imim3i_1"))) ;; A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994.) (Revised by NM, 13-Jul-2013.) (theorem "sylc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylc_1" ("wi" "ph" "ps")) ("sylc_2" ("wi" "ph" "ch")) ("sylc_3" ("wi" "ps" ("wi" "ch" "th")))) (for) ("wi" "ph" "th") ("pm2_43i" "ph" "th" ("syl2im" "ph" "ps" "ph" "ch" "th" "sylc_1" "sylc_2" "sylc_3"))) ;; A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.) (theorem "syl3c" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl3c_1" ("wi" "ph" "ps")) ("syl3c_2" ("wi" "ph" "ch")) ("syl3c_3" ("wi" "ph" "th")) ("syl3c_4" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta"))))) (for) ("wi" "ph" "ta") ("mpd" "ph" "th" "ta" "syl3c_3" ("sylc" "ph" "ps" "ch" ("wi" "th" "ta") "syl3c_1" "syl3c_2" "syl3c_4"))) ;; A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.) (theorem "syl6mpi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl6mpi_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl6mpi_2" "th") ("syl6mpi_3" ("wi" "ch" ("wi" "th" "ta")))) (for) ("wi" "ph" ("wi" "ps" "ta")) ("syl6" "ph" "ps" "ch" "ta" "syl6mpi_1" ("mpi" "ch" "th" "ta" "syl6mpi_2" "syl6mpi_3"))) ;; Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.) (theorem "mpsyl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpsyl_1" "ph") ("mpsyl_2" ("wi" "ps" "ch")) ("mpsyl_3" ("wi" "ph" ("wi" "ch" "th")))) (for) ("wi" "ps" "th") ("sylc" "ps" "ph" "ch" "th" ("a1i" "ph" "ps" "mpsyl_1") "mpsyl_2" "mpsyl_3")) ;; Modus ponens combined with a double syllogism inference. (Contributed by Alan Sare, 22-Jul-2012.) (theorem "mpsylsyld" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mpsylsyld_1" "ph") ("mpsylsyld_2" ("wi" "ps" ("wi" "ch" "th"))) ("mpsylsyld_3" ("wi" "ph" ("wi" "th" "ta")))) (for) ("wi" "ps" ("wi" "ch" "ta")) ("sylsyld" "ps" "ph" "ch" "th" "ta" ("a1i" "ph" "ps" "mpsylsyld_1") "mpsylsyld_2" "mpsylsyld_3")) ;; Inference combining ~ syl6 with contraction. (Contributed by Alan Sare, 2-May-2011.) (theorem "syl6c" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl6c_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl6c_2" ("wi" "ph" ("wi" "ps" "th"))) ("syl6c_3" ("wi" "ch" ("wi" "th" "ta")))) (for) ("wi" "ph" ("wi" "ps" "ta")) ("mpdd" "ph" "ps" "th" "ta" "syl6c_2" ("syl6" "ph" "ps" "ch" ("wi" "th" "ta") "syl6c_1" "syl6c_3"))) ;; A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.) (theorem "syl6ci" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl6ci_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl6ci_2" ("wi" "ph" "th")) ("syl6ci_3" ("wi" "ch" ("wi" "th" "ta")))) (for) ("wi" "ph" ("wi" "ps" "ta")) ("syl6c" "ph" "ps" "ch" "th" "ta" "syl6ci_1" ("a1d" "ph" "th" "ps" "syl6ci_2") "syl6ci_3")) ;; Nested syllogism deduction. Deduction associated with ~ syld . Double deduction associated with ~ syl . (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.) (theorem "syldd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syldd_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th")))) ("syldd_2" ("wi" "ph" ("wi" "ps" ("wi" "th" "ta"))))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "ta"))) ("syl6c" "ph" "ps" ("wi" "th" "ta") ("wi" "ch" "th") ("wi" "ch" "ta") "syldd_2" "syldd_1" ("imim2" "th" "ta" "ch"))) ;; A nested syllogism deduction. Deduction associated with ~ syl5 . (Contributed by NM, 14-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.) (theorem "syl5d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl5d_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl5d_2" ("wi" "ph" ("wi" "th" ("wi" "ch" "ta"))))) (for) ("wi" "ph" ("wi" "th" ("wi" "ps" "ta"))) ("syldd" "ph" "th" "ps" "ch" "ta" ("a1d" "ph" ("wi" "ps" "ch") "th" "syl5d_1") "syl5d_2")) ;; A syllogism rule of inference. The first premise is used to replace the third antecedent of the second premise. (Contributed by NM, 12-Jan-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.) (theorem "syl7" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl7_1" ("wi" "ph" "ps")) ("syl7_2" ("wi" "ch" ("wi" "th" ("wi" "ps" "ta"))))) (for) ("wi" "ch" ("wi" "th" ("wi" "ph" "ta"))) ("syl5d" "ch" "ph" "ps" "th" "ta" ("a1i" ("wi" "ph" "ps") "ch" "syl7_1") "syl7_2")) ;; A nested syllogism deduction. Deduction associated with ~ syl6 . (Contributed by NM, 11-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.) (theorem "syl6d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl6d_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th")))) ("syl6d_2" ("wi" "ph" ("wi" "th" "ta")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "ta"))) ("syldd" "ph" "ps" "ch" "th" "ta" "syl6d_1" ("a1d" "ph" ("wi" "th" "ta") "ps" "syl6d_2"))) ;; A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 3-Aug-2012.) (theorem "syl8" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl8_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th")))) ("syl8_2" ("wi" "th" "ta"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "ta"))) ("syl6d" "ph" "ps" "ch" "th" "ta" "syl8_1" ("a1i" ("wi" "th" "ta") "ph" "syl8_2"))) ;; A nested syllogism inference with different antecedents. (Contributed by NM, 13-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (theorem "syl9" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl9_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl9_2" ("wi" "th" ("wi" "ch" "ta")))) (for) ("wi" "ph" ("wi" "th" ("wi" "ps" "ta"))) ("syl5d" "ph" "ps" "ch" "th" "ta" "syl9_1" ("a1i" ("wi" "th" ("wi" "ch" "ta")) "ph" "syl9_2"))) ;; A nested syllogism inference with different antecedents. (Contributed by NM, 14-May-1993.) (theorem "syl9r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl9r_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl9r_2" ("wi" "th" ("wi" "ch" "ta")))) (for) ("wi" "th" ("wi" "ph" ("wi" "ps" "ta"))) ("com12" "ph" "th" ("wi" "ps" "ta") ("syl9" "ph" "ps" "ch" "th" "ta" "syl9r_1" "syl9r_2"))) ;; A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.) (theorem "syl10" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl10_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl10_2" ("wi" "ph" ("wi" "ps" ("wi" "th" "ta")))) ("syl10_3" ("wi" "ch" ("wi" "ta" "et")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "th" "et"))) ("syldd" "ph" "ps" "th" "ta" "et" "syl10_2" ("syl6" "ph" "ps" "ch" ("wi" "ta" "et") "syl10_1" "syl10_3"))) ;; Triple deduction introducing an antecedent to a wff. Deduction associated with ~ a1dd . Double deduction associated with ~ a1d . Triple deduction associated with ~ ax-1 and ~ a1i . (Contributed by Jeff Hankins, 4-Aug-2009.) (theorem "a1ddd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("a1ddd_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "ta"))))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("syl8" "ph" "ps" "ch" "ta" ("wi" "th" "ta") "a1ddd_1" ("ax_1" "ta" "th"))) ;; Deduction combining antecedents and consequents. Deduction associated with ~ imim12 and ~ imim12i . (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mel L. O'Cat, 30-Oct-2011.) (theorem "imim12d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imim12d_1" ("wi" "ph" ("wi" "ps" "ch"))) ("imim12d_2" ("wi" "ph" ("wi" "th" "ta")))) (for) ("wi" "ph" ("wi" ("wi" "ch" "th") ("wi" "ps" "ta"))) ("syl5d" "ph" "ps" "ch" ("wi" "ch" "th") "ta" "imim12d_1" ("imim2d" "ph" "th" "ta" "ch" "imim12d_2"))) ;; Deduction adding nested consequents. Deduction associated with ~ imim1 and ~ imim1i . (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2012.) (theorem "imim1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("imim1d_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" ("wi" "ch" "th") ("wi" "ps" "th"))) ("imim12d" "ph" "ps" "ch" "th" "th" "imim1d_1" ("idd" "ph" "th"))) ;; A closed form of syllogism (see ~ syl ). Theorem *2.06 of [WhiteheadRussell] p. 100. Its associated inference is ~ imim1i . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.) (theorem "imim1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wi" "ps" "ch") ("wi" "ph" "ch"))) ("imim1d" ("wi" "ph" "ps") "ph" "ps" "ch" ("id" ("wi" "ph" "ps")))) ;; Theorem *2.83 of [WhiteheadRussell] p. 108. Closed form of ~ syld . (Contributed by NM, 3-Jan-2005.) (theorem "pm2_83" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" ("wi" "ph" ("wi" "ch" "th")) ("wi" "ph" ("wi" "ps" "th")))) ("imim3i" ("wi" "ps" "ch") ("wi" "ch" "th") ("wi" "ps" "th") "ph" ("imim1" "ps" "ch" "th"))) ;; Over minimal implicational calculus, Peirce's axiom ~ peirce implies an axiom sometimes called "Roll", ` ( ( ( ph -> ps ) -> ch ) -> ( ( ch -> ph ) -> ph ) ) ` , of which ~ looinv is a special instance. The converse also holds: substitute ` ( ph -> ps ) ` for ` ch ` in Roll and use ~ id and ~ ax-mp . (Contributed by BJ, 15-Jun-2021.) (theorem "peirceroll" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" ("wi" ("wi" "ph" "ps") "ph") "ph") ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" ("wi" "ch" "ph") "ph"))) ("syl5" ("wi" ("wi" "ph" "ps") "ch") ("wi" ("wi" "ch" "ph") ("wi" ("wi" "ph" "ps") "ph")) ("wi" ("wi" ("wi" "ph" "ps") "ph") "ph") ("wi" ("wi" "ch" "ph") "ph") ("imim1" ("wi" "ph" "ps") "ch" "ph") ("imim2" ("wi" ("wi" "ph" "ps") "ph") "ph" ("wi" "ch" "ph")))) ;; Commutation of antecedents. Swap 2nd and 3rd. Deduction associated with ~ com12 . (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.) (theorem "com23" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("com3_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ph" ("wi" "ch" ("wi" "ps" "th"))) ("syl9" "ph" "ps" ("wi" "ch" "th") "ch" "th" "com3_1" ("pm2_27" "ch" "th"))) ;; Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.) (theorem "com3r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("com3_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ch" ("wi" "ph" ("wi" "ps" "th"))) ("com12" "ph" "ch" ("wi" "ps" "th") ("com23" "ph" "ps" "ch" "th" "com3_1"))) ;; Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) (theorem "com13" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("com3_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ch" ("wi" "ps" ("wi" "ph" "th"))) ("com23" "ch" "ph" "ps" "th" ("com3r" "ph" "ps" "ch" "th" "com3_1"))) ;; Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) (theorem "com3l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("com3_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ps" ("wi" "ch" ("wi" "ph" "th"))) ("com3r" "ch" "ph" "ps" "th" ("com3r" "ph" "ps" "ch" "th" "com3_1"))) ;; Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. This was the third axiom in Frege's logic system, specifically Proposition 8 of [Frege1879] p. 35. Its associated inference is ~ com12 . (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Sep-2012.) (theorem "pm2_04" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ps" ("wi" "ph" "ch"))) ("com23" ("wi" "ph" ("wi" "ps" "ch")) "ph" "ps" "ch" ("id" ("wi" "ph" ("wi" "ps" "ch"))))) ;; Commutation of antecedents. Swap 3rd and 4th. Deduction associated with ~ com23 . Double deduction associated with ~ com12 . (Contributed by NM, 25-Apr-1994.) (theorem "com34" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("com4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" "ph" ("wi" "ps" ("wi" "th" ("wi" "ch" "ta")))) ("syl6" "ph" "ps" ("wi" "ch" ("wi" "th" "ta")) ("wi" "th" ("wi" "ch" "ta")) "com4_1" ("pm2_04" "ch" "th" "ta"))) ;; Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Mel L. O'Cat, 15-Aug-2004.) (theorem "com4l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("com4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ph" "ta")))) ("com34" "ps" "ch" "ph" "th" "ta" ("com3l" "ph" "ps" "ch" ("wi" "th" "ta") "com4_1"))) ;; Commutation of antecedents. Rotate twice. (Contributed by NM, 25-Apr-1994.) (theorem "com4t" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("com4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" "ch" ("wi" "th" ("wi" "ph" ("wi" "ps" "ta")))) ("com4l" "ps" "ch" "th" "ph" "ta" ("com4l" "ph" "ps" "ch" "th" "ta" "com4_1"))) ;; Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.) (theorem "com4r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("com4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" "th" ("wi" "ph" ("wi" "ps" ("wi" "ch" "ta")))) ("com4l" "ch" "th" "ph" "ps" "ta" ("com4t" "ph" "ps" "ch" "th" "ta" "com4_1"))) ;; Commutation of antecedents. Swap 2nd and 4th. Deduction associated with ~ com13 . (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) (theorem "com24" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("com4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" "ph" ("wi" "th" ("wi" "ch" ("wi" "ps" "ta")))) ("com13" "ch" "th" "ph" ("wi" "ps" "ta") ("com4t" "ph" "ps" "ch" "th" "ta" "com4_1"))) ;; Commutation of antecedents. Swap 1st and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) (theorem "com14" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("com4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" "th" ("wi" "ps" ("wi" "ch" ("wi" "ph" "ta")))) ("com3r" "ps" "ch" "th" ("wi" "ph" "ta") ("com4l" "ph" "ps" "ch" "th" "ta" "com4_1"))) ;; Commutation of antecedents. Swap 4th and 5th. Deduction associated with ~ com34 . Double deduction associated with ~ com23 . Triple deduction associated with ~ com12 . (Contributed by Jeff Hankins, 28-Jun-2009.) (theorem "com45" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("com5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "ta" ("wi" "th" "et"))))) ("syl8" "ph" "ps" "ch" ("wi" "th" ("wi" "ta" "et")) ("wi" "ta" ("wi" "th" "et")) "com5_1" ("pm2_04" "th" "ta" "et"))) ;; Commutation of antecedents. Swap 3rd and 5th. Deduction associated with ~ com24 . Double deduction associated with ~ com13 . (Contributed by Jeff Hankins, 28-Jun-2009.) (theorem "com35" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("com5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ta" ("wi" "th" ("wi" "ch" "et"))))) ("com34" "ph" "ps" "th" "ta" ("wi" "ch" "et") ("com45" "ph" "ps" "th" "ch" "ta" "et" ("com34" "ph" "ps" "ch" "th" ("wi" "ta" "et") "com5_1")))) ;; Commutation of antecedents. Swap 2nd and 5th. Deduction associated with ~ com14 . (Contributed by Jeff Hankins, 28-Jun-2009.) (theorem "com25" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("com5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" "ph" ("wi" "ta" ("wi" "ch" ("wi" "th" ("wi" "ps" "et"))))) ("com24" "ph" "th" "ch" "ta" ("wi" "ps" "et") ("com45" "ph" "th" "ch" "ps" "ta" "et" ("com24" "ph" "ps" "ch" "th" ("wi" "ta" "et") "com5_1")))) ;; Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.) (theorem "com5l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("com5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" ("wi" "ph" "et"))))) ("com45" "ps" "ch" "th" "ph" "ta" "et" ("com4l" "ph" "ps" "ch" "th" ("wi" "ta" "et") "com5_1"))) ;; Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.) (theorem "com15" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("com5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" "ta" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ph" "et"))))) ("com4r" "ps" "ch" "th" "ta" ("wi" "ph" "et") ("com5l" "ph" "ps" "ch" "th" "ta" "et" "com5_1"))) ;; Commutation of antecedents. Rotate left twice. (Contributed by Jeff Hankins, 28-Jun-2009.) (theorem "com52l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("com5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" "ch" ("wi" "th" ("wi" "ta" ("wi" "ph" ("wi" "ps" "et"))))) ("com5l" "ps" "ch" "th" "ta" "ph" "et" ("com5l" "ph" "ps" "ch" "th" "ta" "et" "com5_1"))) ;; Commutation of antecedents. Rotate right twice. (Contributed by Jeff Hankins, 28-Jun-2009.) (theorem "com52r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("com5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" "th" ("wi" "ta" ("wi" "ph" ("wi" "ps" ("wi" "ch" "et"))))) ("com5l" "ch" "th" "ta" "ph" "ps" "et" ("com52l" "ph" "ps" "ch" "th" "ta" "et" "com5_1"))) ;; Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen, 29-Jul-2012.) (theorem "com5r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("com5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" "ta" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "et"))))) ("com52l" "ch" "th" "ta" "ph" "ps" "et" ("com52l" "ph" "ps" "ch" "th" "ta" "et" "com5_1"))) ;; Closed form of ~ imim12i and of ~ 3syl . (Contributed by BJ, 16-Jul-2019.) (theorem "imim12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wi" "ch" "th") ("wi" ("wi" "ps" "ch") ("wi" "ph" "th")))) ("com24" ("wi" "ph" "ps") "ph" ("wi" "ps" "ch") ("wi" "ch" "th") "th" ("imim2i" "ps" ("wi" ("wi" "ps" "ch") ("wi" ("wi" "ch" "th") "th")) "ph" ("com13" ("wi" "ch" "th") ("wi" "ps" "ch") "ps" "th" ("imim2" "ch" "th" "ps"))))) ;; Elimination of a nested antecedent as a partial converse of ~ ja (the other being ~ jarl ). (Contributed by Wolf Lammen, 9-May-2013.) (theorem "jarr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" "ps" "ch")) ("imim1i" "ps" ("wi" "ph" "ps") "ch" ("ax_1" "ps" "ph"))) ;; Deduction associated with ~ pm2.86 . (Contributed by NM, 29-Jun-1995.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) (theorem "pm2_86d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm2_86d_1" ("wi" "ph" ("wi" ("wi" "ps" "ch") ("wi" "ps" "th"))))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))) ("com23" "ph" "ch" "ps" "th" ("syl5" "ch" ("wi" "ps" "ch") "ph" ("wi" "ps" "th") ("ax_1" "ch" "ps") "pm2_86d_1"))) ;; Converse of axiom ~ ax-2 . Theorem *2.86 of [WhiteheadRussell] p. 108. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) (theorem "pm2_86" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("wi" "ph" ("wi" "ps" "ch"))) ("pm2_86d" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch")) "ph" "ps" "ch" ("id" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch"))))) ;; Inference associated with ~ pm2.86 . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) (theorem "pm2_86i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_86i_1" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch")))) (for) ("wi" "ph" ("wi" "ps" "ch")) ("syl11" ("wi" "ph" "ps") "ph" "ch" "ps" "pm2_86i_1" ("ax_1" "ps" "ph"))) ;; The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. See ~ loowoz for an alternate axiom. (Contributed by Mel L. O'Cat, 12-Aug-2004.) (theorem "loolin" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ps" "ph")) ("wi" "ps" "ph")) ("pm2_43d" ("wi" ("wi" "ph" "ps") ("wi" "ps" "ph")) "ps" "ph" ("jarr" "ph" "ps" ("wi" "ps" "ph")))) ;; An alternate for the Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz ~ loolin , due to Barbara Wozniakowska, _Reports on Mathematical Logic_ 10, 129-137 (1978). (Contributed by Mel L. O'Cat, 8-Aug-2004.) (theorem "loowoz" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("wi" ("wi" "ps" "ph") ("wi" "ps" "ch"))) ("a2d" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch")) "ps" "ph" "ch" ("jarr" "ph" "ps" ("wi" "ph" "ch")))) ;; Alias for ~ ax-3 to be used instead of it for labeling consistency. Its associated inference is ~ con4i and its associated deduction is ~ con4d . (Contributed by BJ, 24-Dec-2020.) (theorem "con4" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" ("wn" "ph") ("wn" "ps")) ("wi" "ps" "ph")) ("ax_3" "ph" "ps")) ;; Inference associated with ~ con4 . Its associated inference is ~ mt4 . Remark: this can also be proved using ~ notnot followed by ~ nsyl2 , giving a shorter proof but depending on more axioms (namely, ~ ax-1 and ~ ax-2 ). (Contributed by NM, 29-Dec-1992.) (theorem "con4i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("con4i_1" ("wi" ("wn" "ph") ("wn" "ps")))) (for) ("wi" "ps" "ph") ("ax_mp" ("wi" ("wn" "ph") ("wn" "ps")) ("wi" "ps" "ph") "con4i_1" ("con4" "ph" "ps"))) ;; Deduction associated with ~ con4 . (Contributed by NM, 26-Mar-1995.) (theorem "con4d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("con4d_1" ("wi" "ph" ("wi" ("wn" "ps") ("wn" "ch"))))) (for) ("wi" "ph" ("wi" "ch" "ps")) ("syl" "ph" ("wi" ("wn" "ps") ("wn" "ch")) ("wi" "ch" "ps") "con4d_1" ("con4" "ps" "ch"))) ;; The rule of modus tollens. Inference associated with ~ con4i . (Contributed by Wolf Lammen, 12-May-2013.) (theorem "mt4" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("mt4_1" "ph") ("mt4_2" ("wi" ("wn" "ps") ("wn" "ph")))) (for) "ps" ("ax_mp" "ph" "ps" "mt4_1" ("con4i" "ps" "ph" "mt4_2"))) ;; A contradiction implies anything. Inference associated with ~ pm2.21 . Its associated inference is ~ pm2.24ii . (Contributed by NM, 16-Sep-1993.) (theorem "pm2_21i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm2_21i_1" ("wn" "ph"))) (for) ("wi" "ph" "ps") ("con4i" "ps" "ph" ("a1i" ("wn" "ph") ("wn" "ps") "pm2_21i_1"))) ;; A contradiction implies anything. Inference associated with ~ pm2.21i and ~ pm2.24i . (Contributed by NM, 27-Feb-2008.) (theorem "pm2_24ii" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm2_24ii_1" "ph") ("pm2_24ii_2" ("wn" "ph"))) (for) "ps" ("ax_mp" "ph" "ps" "pm2_24ii_1" ("pm2_21i" "ph" "ps" "pm2_24ii_2"))) ;; A contradiction implies anything. Deduction associated with ~ pm2.21 . (Contributed by NM, 10-Feb-1996.) (theorem "pm2_21d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_21d_1" ("wi" "ph" ("wn" "ps")))) (for) ("wi" "ph" ("wi" "ps" "ch")) ("con4d" "ph" "ch" "ps" ("a1d" "ph" ("wn" "ps") ("wn" "ch") "pm2_21d_1"))) ;; Alternate proof of ~ pm2.21dd . (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "pm2_21ddALT" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_21ddALT_1" ("wi" "ph" "ps")) ("pm2_21ddALT_2" ("wi" "ph" ("wn" "ps")))) (for) ("wi" "ph" "ch") ("mpd" "ph" "ps" "ch" "pm2_21ddALT_1" ("pm2_21d" "ph" "ps" "ch" "pm2_21ddALT_2"))) ;; From a wff and its negation, anything is true. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. Its associated inference is ~ pm2.21i . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 14-Sep-2012.) (theorem "pm2_21" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wi" "ph" "ps")) ("pm2_21d" ("wn" "ph") "ph" "ps" ("id" ("wn" "ph")))) ;; Theorem *2.24 of [WhiteheadRussell] p. 104. Its associated inference is ~ pm2.24i . (Contributed by NM, 3-Jan-2005.) (theorem "pm2_24" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wi" ("wn" "ph") "ps")) ("com12" ("wn" "ph") "ph" "ps" ("pm2_21" "ph" "ps"))) ;; Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also called the Law of Clavius. See also ~ pm2.01 . (Contributed by NM, 29-Dec-1992.) (theorem "pm2_18" (for ("ph" ( "wff"))) (for) (for) ("wi" ("wi" ("wn" "ph") "ph") "ph") ("pm2_43i" ("wi" ("wn" "ph") "ph") "ph" ("con4d" ("wi" ("wn" "ph") "ph") "ph" ("wi" ("wn" "ph") "ph") ("a2i" ("wn" "ph") "ph" ("wn" ("wi" ("wn" "ph") "ph")) ("pm2_21" "ph" ("wn" ("wi" ("wn" "ph") "ph"))))))) ;; Inference associated with ~ pm2.18 . (Contributed by BJ, 30-Mar-2020.) (theorem "pm2_18i" (for ("ph" ( "wff"))) (for ("pm2_18i_1" ("wi" ("wn" "ph") "ph"))) (for) "ph" ("ax_mp" ("wi" ("wn" "ph") "ph") "ph" "pm2_18i_1" ("pm2_18" "ph"))) ;; Deduction based on reductio ad absurdum. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "pm2_18d" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm2_18d_1" ("wi" "ph" ("wi" ("wn" "ps") "ps")))) (for) ("wi" "ph" "ps") ("syl" "ph" ("wi" ("wn" "ps") "ps") "ps" "pm2_18d_1" ("pm2_18" "ps"))) ;; Double negation elimination. Converse of ~ notnot and one implication of ~ notnotb . Theorem *2.14 of [WhiteheadRussell] p. 102. This was the fifth axiom of Frege, specifically Proposition 31 of [Frege1879] p. 44. In classical logic (our logic) this is always true. In intuitionistic logic this is not always true, and formulas for which it is true are called "stable." (Contributed by NM, 29-Dec-1992.) (Proof shortened by David Harvey, 5-Sep-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (theorem "notnotr" (for ("ph" ( "wff"))) (for) (for) ("wi" ("wn" ("wn" "ph")) "ph") ("pm2_18d" ("wn" ("wn" "ph")) "ph" ("pm2_21" ("wn" "ph") "ph"))) ;; Inference associated with ~ notnotr . Remark: the proof via ~ notnotr and ~ ax-mp also has three essential steps, but has a total number of steps equal to 8, instead of the present 7, because it has to construct the formula ` ph ` twice and the formula ` -. -. ph ` , whereas the present proof has to construct the formula ` ph ` twice and the formula ` -. ph ` , and therefore makes only one use of ~ wn instead of two. This can be checked by running the Metamath command "SHOW PROOF notnotri / NORMAL". (Contributed by NM, 27-Feb-2008.) (Proof shortened by Wolf Lammen, 15-Jul-2021.) (theorem "notnotri" (for ("ph" ( "wff"))) (for ("notnotri_1" ("wn" ("wn" "ph")))) (for) "ph" ("pm2_18i" "ph" ("pm2_21i" ("wn" "ph") "ph" "notnotri_1"))) ;; Obsolete proof of ~ notnotri as of 15-Jul-2021 . (Contributed by NM, 27-Feb-2008.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "notnotriOLD" (for ("ph" ( "wff"))) (for ("notnotri_1" ("wn" ("wn" "ph")))) (for) "ph" ("ax_mp" ("wn" ("wn" "ph")) "ph" "notnotri_1" ("notnotr" "ph"))) ;; Deduction associated with ~ notnotr and ~ notnotri . Double negation elimination rule. A translation of the natural deduction rule ` -. -. ` C , ` _G |- -. -. ps => _G |- ps ` ; see ~ natded . This is Definition NNC in [Pfenning] p. 17. This rule is valid in classical logic (our logic), but not in intuitionistic logic. (Contributed by DAW, 8-Feb-2017.) (theorem "notnotrd" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("notnotrd_1" ("wi" "ph" ("wn" ("wn" "ps"))))) (for) ("wi" "ph" "ps") ("syl" "ph" ("wn" ("wn" "ps")) "ps" "notnotrd_1" ("notnotr" "ps"))) ;; A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (theorem "con2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("con2d_1" ("wi" "ph" ("wi" "ps" ("wn" "ch"))))) (for) ("wi" "ph" ("wi" "ch" ("wn" "ps"))) ("con4d" "ph" ("wn" "ps") "ch" ("syl5" ("wn" ("wn" "ps")) "ps" "ph" ("wn" "ch") ("notnotr" "ps") "con2d_1"))) ;; Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Feb-2013.) (theorem "con2" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" "ph" ("wn" "ps")) ("wi" "ps" ("wn" "ph"))) ("con2d" ("wi" "ph" ("wn" "ps")) "ph" "ps" ("id" ("wi" "ph" ("wn" "ps"))))) ;; Modus tollens deduction. (Contributed by NM, 4-Jul-1994.) (theorem "mt2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mt2d_1" ("wi" "ph" "ch")) ("mt2d_2" ("wi" "ph" ("wi" "ps" ("wn" "ch"))))) (for) ("wi" "ph" ("wn" "ps")) ("mpd" "ph" "ch" ("wn" "ps") "mt2d_1" ("con2d" "ph" "ps" "ch" "mt2d_2"))) ;; Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.) (theorem "mt2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mt2i_1" "ch") ("mt2i_2" ("wi" "ph" ("wi" "ps" ("wn" "ch"))))) (for) ("wi" "ph" ("wn" "ps")) ("mt2d" "ph" "ps" "ch" ("a1i" "ch" "ph" "mt2i_1") "mt2i_2")) ;; A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (theorem "nsyl3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("nsyl3_1" ("wi" "ph" ("wn" "ps"))) ("nsyl3_2" ("wi" "ch" "ps"))) (for) ("wi" "ch" ("wn" "ph")) ("mt2d" "ch" "ph" "ps" "nsyl3_2" ("a1i" ("wi" "ph" ("wn" "ps")) "ch" "nsyl3_1"))) ;; A contraposition inference. Its associated inference is ~ mt2 . (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 13-Jun-2013.) (theorem "con2i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("con2i_a" ("wi" "ph" ("wn" "ps")))) (for) ("wi" "ps" ("wn" "ph")) ("nsyl3" "ph" "ps" "ps" "con2i_a" ("id" "ps"))) ;; A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.) (theorem "nsyl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("nsyl_1" ("wi" "ph" ("wn" "ps"))) ("nsyl_2" ("wi" "ch" "ps"))) (for) ("wi" "ph" ("wn" "ch")) ("con2i" "ch" "ph" ("nsyl3" "ph" "ps" "ch" "nsyl_1" "nsyl_2"))) ;; Double negation introduction. Converse of ~ notnotr and one implication of ~ notnotb . Theorem *2.12 of [WhiteheadRussell] p. 101. This was the sixth axiom of Frege, specifically Proposition 41 of [Frege1879] p. 47. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.) (theorem "notnot" (for ("ph" ( "wff"))) (for) (for) ("wi" "ph" ("wn" ("wn" "ph"))) ("con2i" ("wn" "ph") "ph" ("id" ("wn" "ph")))) ;; Inference associated with ~ notnot . (Contributed by NM, 27-Feb-2008.) (theorem "notnoti" (for ("ph" ( "wff"))) (for ("notnoti_1" "ph")) (for) ("wn" ("wn" "ph")) ("ax_mp" "ph" ("wn" ("wn" "ph")) "notnoti_1" ("notnot" "ph"))) ;; Deduction associated with ~ notnot and ~ notnoti . (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.) (theorem "notnotd" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("notnotd_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("wn" ("wn" "ps"))) ("syl" "ph" "ps" ("wn" ("wn" "ps")) "notnotd_1" ("notnot" "ps"))) ;; A contraposition deduction. (Contributed by NM, 27-Dec-1992.) (theorem "con1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("con1d_1" ("wi" "ph" ("wi" ("wn" "ps") "ch")))) (for) ("wi" "ph" ("wi" ("wn" "ch") "ps")) ("con4d" "ph" "ps" ("wn" "ch") ("syl6" "ph" ("wn" "ps") "ch" ("wn" ("wn" "ch")) "con1d_1" ("notnot" "ch")))) ;; Modus tollens deduction. (Contributed by NM, 26-Mar-1995.) (theorem "mt3d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mt3d_1" ("wi" "ph" ("wn" "ch"))) ("mt3d_2" ("wi" "ph" ("wi" ("wn" "ps") "ch")))) (for) ("wi" "ph" "ps") ("mpd" "ph" ("wn" "ch") "ps" "mt3d_1" ("con1d" "ph" "ps" "ch" "mt3d_2"))) ;; Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.) (theorem "mt3i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mt3i_1" ("wn" "ch")) ("mt3i_2" ("wi" "ph" ("wi" ("wn" "ps") "ch")))) (for) ("wi" "ph" "ps") ("mt3d" "ph" "ps" "ch" ("a1i" ("wn" "ch") "ph" "mt3i_1") "mt3i_2")) ;; A negated syllogism inference. (Contributed by NM, 26-Jun-1994.) (theorem "nsyl2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("nsyl2_1" ("wi" "ph" ("wn" "ps"))) ("nsyl2_2" ("wi" ("wn" "ch") "ps"))) (for) ("wi" "ph" "ch") ("mt3d" "ph" "ch" "ps" "nsyl2_1" ("a1i" ("wi" ("wn" "ch") "ps") "ph" "nsyl2_2"))) ;; Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. Its associated inference is ~ con1i . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Feb-2013.) (theorem "con1" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" ("wn" "ph") "ps") ("wi" ("wn" "ps") "ph")) ("con1d" ("wi" ("wn" "ph") "ps") "ph" "ps" ("id" ("wi" ("wn" "ph") "ps")))) ;; A contraposition inference. Inference associated with ~ con1 . Its associated inference is ~ mt3 . (Contributed by NM, 3-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Jun-2013.) (theorem "con1i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("con1i_1" ("wi" ("wn" "ph") "ps"))) (for) ("wi" ("wn" "ps") "ph") ("nsyl2" ("wn" "ps") "ps" "ph" ("id" ("wn" "ps")) "con1i_1")) ;; Obsolete proof of ~ con4i as of 15-Jul-2021. This shorter proof has been reverted to its original to avoid a dependency on ~ ax-1 and ~ ax-2 . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 21-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "con4iOLD" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("con4iOLD_1" ("wi" ("wn" "ph") ("wn" "ps")))) (for) ("wi" "ps" "ph") ("nsyl2" "ps" ("wn" "ps") "ph" ("notnot" "ps") "con4iOLD_1")) ;; Inference associated with ~ pm2.24 . Its associated inference is ~ pm2.24ii . (Contributed by NM, 20-Aug-2001.) (theorem "pm2_24i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm2_24i_1" "ph")) (for) ("wi" ("wn" "ph") "ps") ("con1i" "ps" "ph" ("a1i" "ph" ("wn" "ps") "pm2_24i_1"))) ;; Deduction form of ~ pm2.24 . (Contributed by NM, 30-Jan-2006.) (theorem "pm2_24d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_24d_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("wi" ("wn" "ps") "ch")) ("con1d" "ph" "ch" "ps" ("a1d" "ph" "ps" ("wn" "ch") "pm2_24d_1"))) ;; A contraposition deduction. Deduction form of ~ con3 . (Contributed by NM, 10-Jan-1993.) (theorem "con3d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("con3d_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" ("wn" "ch") ("wn" "ps"))) ("con1d" "ph" ("wn" "ps") "ch" ("syl5" ("wn" ("wn" "ps")) "ps" "ph" "ch" ("notnotr" "ps") "con3d_1"))) ;; Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This was the fourth axiom of Frege, specifically Proposition 28 of [Frege1879] p. 43. Its associated inference is ~ con3i . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 13-Feb-2013.) (theorem "con3" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wn" "ps") ("wn" "ph"))) ("con3d" ("wi" "ph" "ps") "ph" "ps" ("id" ("wi" "ph" "ps")))) ;; A contraposition inference. Inference associated with ~ con3 . Its associated inference is ~ mto . (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.) (theorem "con3i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("con3i_a" ("wi" "ph" "ps"))) (for) ("wi" ("wn" "ps") ("wn" "ph")) ("nsyl" ("wn" "ps") "ps" "ph" ("id" ("wn" "ps")) "con3i_a")) ;; Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.) (theorem "con3rr3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("con3rr3_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" ("wn" "ch") ("wi" "ph" ("wn" "ps"))) ("com12" "ph" ("wn" "ch") ("wn" "ps") ("con3d" "ph" "ps" "ch" "con3rr3_1"))) ;; Modus tollens deduction. Deduction form of ~ mt4 . (Contributed by NM, 9-Jun-2006.) (theorem "mt4d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mt4d_1" ("wi" "ph" "ps")) ("mt4d_2" ("wi" "ph" ("wi" ("wn" "ch") ("wn" "ps"))))) (for) ("wi" "ph" "ch") ("mpd" "ph" "ps" "ch" "mt4d_1" ("con4d" "ph" "ch" "ps" "mt4d_2"))) ;; Modus tollens inference. (Contributed by Wolf Lammen, 12-May-2013.) (theorem "mt4i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mt4i_1" "ch") ("mt4i_2" ("wi" "ph" ("wi" ("wn" "ps") ("wn" "ch"))))) (for) ("wi" "ph" "ps") ("mt4d" "ph" "ch" "ps" ("a1i" "ch" "ph" "mt4i_1") "mt4i_2")) ;; A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.) (theorem "nsyld" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("ta" ( "wff"))) (for ("nsyld_1" ("wi" "ph" ("wi" "ps" ("wn" "ch")))) ("nsyld_2" ("wi" "ph" ("wi" "ta" "ch")))) (for) ("wi" "ph" ("wi" "ps" ("wn" "ta"))) ("syld" "ph" "ps" ("wn" "ch") ("wn" "ta") "nsyld_1" ("con3d" "ph" "ta" "ch" "nsyld_2"))) ;; A negated syllogism inference. (Contributed by NM, 3-May-1994.) (theorem "nsyli" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("nsyli_1" ("wi" "ph" ("wi" "ps" "ch"))) ("nsyli_2" ("wi" "th" ("wn" "ch")))) (for) ("wi" "ph" ("wi" "th" ("wn" "ps"))) ("syl5" "th" ("wn" "ch") "ph" ("wn" "ps") "nsyli_2" ("con3d" "ph" "ps" "ch" "nsyli_1"))) ;; A negated syllogism inference. (Contributed by NM, 15-Feb-1996.) (theorem "nsyl4" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("nsyl4_1" ("wi" "ph" "ps")) ("nsyl4_2" ("wi" ("wn" "ph") "ch"))) (for) ("wi" ("wn" "ch") "ps") ("syl" ("wn" "ch") "ph" "ps" ("con1i" "ph" "ch" "nsyl4_2") "nsyl4_1")) ;; Theorem *3.2 of [WhiteheadRussell] p. 111, expressed with primitive connectives (see ~ pm3.2 ). (Contributed by NM, 29-Dec-1992.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (theorem "pm3_2im" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wi" "ps" ("wn" ("wi" "ph" ("wn" "ps"))))) ("con2d" "ph" ("wi" "ph" ("wn" "ps")) "ps" ("pm2_27" "ph" ("wn" "ps")))) ;; Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (theorem "mth8" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wi" ("wn" "ps") ("wn" ("wi" "ph" "ps")))) ("con3d" "ph" ("wi" "ph" "ps") "ps" ("pm2_27" "ph" "ps"))) ;; Deduction joining the consequents of two premises. A deduction associated with ~ pm3.2im . (Contributed by NM, 28-Dec-1992.) (theorem "jc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("jc_1" ("wi" "ph" "ps")) ("jc_2" ("wi" "ph" "ch"))) (for) ("wi" "ph" ("wn" ("wi" "ps" ("wn" "ch")))) ("sylc" "ph" "ps" "ch" ("wn" ("wi" "ps" ("wn" "ch"))) "jc_1" "jc_2" ("pm3_2im" "ps" "ch"))) ;; An importation inference. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 20-Jul-2013.) (theorem "impi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("impi_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" ("wn" ("wi" "ph" ("wn" "ps"))) "ch") ("con1i" "ch" ("wi" "ph" ("wn" "ps")) ("con3rr3" "ph" "ps" "ch" "impi_1"))) ;; An exportation inference. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (theorem "expi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("expi_1" ("wi" ("wn" ("wi" "ph" ("wn" "ps"))) "ch"))) (for) ("wi" "ph" ("wi" "ps" "ch")) ("syl6" "ph" "ps" ("wn" ("wi" "ph" ("wn" "ps"))) "ch" ("pm3_2im" "ph" "ps") "expi_1")) ;; Simplification. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) (theorem "simprim" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wi" "ph" ("wn" "ps"))) "ps") ("impi" "ph" "ps" "ps" ("idd" "ph" "ps"))) ;; Simplification. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 21-Jul-2012.) (theorem "simplim" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wi" "ph" "ps")) "ph") ("con1i" "ph" ("wi" "ph" "ps") ("pm2_21" "ph" "ps"))) ;; Theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 9-Oct-2012.) (theorem "pm2_5" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wi" "ph" "ps")) ("wi" ("wn" "ph") "ps")) ("pm2_24d" ("wn" ("wi" "ph" "ps")) "ph" "ps" ("simplim" "ph" "ps"))) ;; Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_51" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wi" "ph" "ps")) ("wi" "ph" ("wn" "ps"))) ("a1d" ("wn" ("wi" "ph" "ps")) ("wn" "ps") "ph" ("con3i" "ps" ("wi" "ph" "ps") ("ax_1" "ps" "ph")))) ;; Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.) (theorem "pm2_521" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wi" "ph" "ps")) ("wi" "ps" "ph")) ("a1d" ("wn" ("wi" "ph" "ps")) "ph" "ps" ("simplim" "ph" "ps"))) ;; Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.) (theorem "pm2_52" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wi" "ph" "ps")) ("wi" ("wn" "ph") ("wn" "ps"))) ("con3d" ("wn" ("wi" "ph" "ps")) "ps" "ph" ("pm2_521" "ph" "ps"))) ;; Exportation theorem ~ ex expressed with primitive connectives. (Contributed by NM, 28-Dec-1992.) (theorem "expt" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" ("wn" ("wi" "ph" ("wn" "ps"))) "ch") ("wi" "ph" ("wi" "ps" "ch"))) ("com12" "ph" ("wi" ("wn" ("wi" "ph" ("wn" "ps"))) "ch") ("wi" "ps" "ch") ("imim1d" "ph" "ps" ("wn" ("wi" "ph" ("wn" "ps"))) "ch" ("pm3_2im" "ph" "ps")))) ;; Importation theorem ~ imp expressed with primitive connectives. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jul-2013.) (theorem "impt" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" ("wn" ("wi" "ph" ("wn" "ps"))) "ch")) ("mpdi" ("wi" "ph" ("wi" "ps" "ch")) ("wn" ("wi" "ph" ("wn" "ps"))) "ps" "ch" ("simprim" "ph" "ps") ("imim1i" ("wn" ("wi" "ph" ("wn" "ps"))) "ph" ("wi" "ps" "ch") ("simplim" "ph" ("wn" "ps"))))) ;; Deduction eliminating an antecedent. (Contributed by NM, 27-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.) (theorem "pm2_61d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_61d_1" ("wi" "ph" ("wi" "ps" "ch"))) ("pm2_61d_2" ("wi" "ph" ("wi" ("wn" "ps") "ch")))) (for) ("wi" "ph" "ch") ("pm2_18d" "ph" "ch" ("syld" "ph" ("wn" "ch") "ps" "ch" ("con1d" "ph" "ps" "ch" "pm2_61d_2") "pm2_61d_1"))) ;; Inference eliminating an antecedent. (Contributed by NM, 15-Jul-2005.) (theorem "pm2_61d1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_61d1_1" ("wi" "ph" ("wi" "ps" "ch"))) ("pm2_61d1_2" ("wi" ("wn" "ps") "ch"))) (for) ("wi" "ph" "ch") ("pm2_61d" "ph" "ps" "ch" "pm2_61d1_1" ("a1i" ("wi" ("wn" "ps") "ch") "ph" "pm2_61d1_2"))) ;; Inference eliminating an antecedent. (Contributed by NM, 18-Aug-1993.) (theorem "pm2_61d2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_61d2_1" ("wi" "ph" ("wi" ("wn" "ps") "ch"))) ("pm2_61d2_2" ("wi" "ps" "ch"))) (for) ("wi" "ph" "ch") ("pm2_61d" "ph" "ps" "ch" ("a1i" ("wi" "ps" "ch") "ph" "pm2_61d2_2") "pm2_61d2_1")) ;; Inference joining the antecedents of two premises. For partial converses, see ~ jarr and ~ jarl . (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) (theorem "ja" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ja_1" ("wi" ("wn" "ph") "ch")) ("ja_2" ("wi" "ps" "ch"))) (for) ("wi" ("wi" "ph" "ps") "ch") ("pm2_61d1" ("wi" "ph" "ps") "ph" "ch" ("imim2i" "ps" "ch" "ph" "ja_2") "ja_1")) ;; Deduction form of ~ ja . (Contributed by Scott Fenton, 13-Dec-2010.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (theorem "jad" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jad_1" ("wi" "ph" ("wi" ("wn" "ps") "th"))) ("jad_2" ("wi" "ph" ("wi" "ch" "th")))) (for) ("wi" "ph" ("wi" ("wi" "ps" "ch") "th")) ("com12" ("wi" "ps" "ch") "ph" "th" ("ja" "ps" "ch" ("wi" "ph" "th") ("com12" "ph" ("wn" "ps") "th" "jad_1") ("com12" "ph" "ch" "th" "jad_2")))) ;; Elimination of a nested antecedent as a partial converse of ~ ja (the other being ~ jarr ). (Contributed by Wolf Lammen, 10-May-2013.) (theorem "jarl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" ("wn" "ph") "ch")) ("imim1i" ("wn" "ph") ("wi" "ph" "ps") "ch" ("pm2_21" "ph" "ps"))) ;; Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.) (theorem "pm2_61i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm2_61i_1" ("wi" "ph" "ps")) ("pm2_61i_2" ("wi" ("wn" "ph") "ps"))) (for) "ps" ("ax_mp" ("wi" "ph" "ph") "ps" ("id" "ph") ("ja" "ph" "ph" "ps" "pm2_61i_2" "pm2_61i_1"))) ;; Inference eliminating two antecedents. (Contributed by NM, 4-Jan-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (theorem "pm2_61ii" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_61ii_1" ("wi" ("wn" "ph") ("wi" ("wn" "ps") "ch"))) ("pm2_61ii_2" ("wi" "ph" "ch")) ("pm2_61ii_3" ("wi" "ps" "ch"))) (for) "ch" ("pm2_61i" "ph" "ch" "pm2_61ii_2" ("pm2_61d2" ("wn" "ph") "ps" "ch" "pm2_61ii_1" "pm2_61ii_3"))) ;; Inference eliminating two antecedents. (Contributed by NM, 13-Jul-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) (theorem "pm2_61nii" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_61nii_1" ("wi" "ph" ("wi" "ps" "ch"))) ("pm2_61nii_2" ("wi" ("wn" "ph") "ch")) ("pm2_61nii_3" ("wi" ("wn" "ps") "ch"))) (for) "ch" ("pm2_61i" "ph" "ch" ("pm2_61d1" "ph" "ps" "ch" "pm2_61nii_1" "pm2_61nii_3") "pm2_61nii_2")) ;; Inference eliminating three antecedents. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) (theorem "pm2_61iii" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm2_61iii_1" ("wi" ("wn" "ph") ("wi" ("wn" "ps") ("wi" ("wn" "ch") "th")))) ("pm2_61iii_2" ("wi" "ph" "th")) ("pm2_61iii_3" ("wi" "ps" "th")) ("pm2_61iii_4" ("wi" "ch" "th"))) (for) "th" ("pm2_61i" "ch" "th" "pm2_61iii_4" ("pm2_61ii" "ph" "ps" ("wi" ("wn" "ch") "th") "pm2_61iii_1" ("a1d" "ph" "th" ("wn" "ch") "pm2_61iii_2") ("a1d" "ps" "th" ("wn" "ch") "pm2_61iii_3")))) ;; Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. Also called the weak Clavius law, and provable in minimal calculus, contrary to the Clavius law ~ pm2.18 . (Contributed by NM, 18-Aug-1993.) (Proof shortened by Mel L. O'Cat, 21-Nov-2008.) (Proof shortened by Wolf Lammen, 31-Oct-2012.) (theorem "pm2_01" (for ("ph" ( "wff"))) (for) (for) ("wi" ("wi" "ph" ("wn" "ph")) ("wn" "ph")) ("ja" "ph" ("wn" "ph") ("wn" "ph") ("id" ("wn" "ph")) ("id" ("wn" "ph")))) ;; Deduction based on reductio ad absurdum. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Mar-2013.) (theorem "pm2_01d" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm2_01d_1" ("wi" "ph" ("wi" "ps" ("wn" "ps"))))) (for) ("wi" "ph" ("wn" "ps")) ("pm2_61d1" "ph" "ps" ("wn" "ps") "pm2_01d_1" ("id" ("wn" "ps")))) ;; Theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_6" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" ("wn" "ph") "ps") ("wi" ("wi" "ph" "ps") "ps")) ("jad" ("wi" ("wn" "ph") "ps") "ph" "ps" "ps" ("id" ("wi" ("wn" "ph") "ps")) ("idd" ("wi" ("wn" "ph") "ps") "ps"))) ;; Theorem *2.61 of [WhiteheadRussell] p. 107. Useful for eliminating an antecedent. (Contributed by NM, 4-Jan-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) (theorem "pm2_61" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wi" ("wn" "ph") "ps") "ps")) ("com12" ("wi" ("wn" "ph") "ps") ("wi" "ph" "ps") "ps" ("pm2_6" "ph" "ps"))) ;; Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.) (theorem "pm2_65" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wi" "ph" ("wn" "ps")) ("wn" "ph"))) ("jad" ("wi" "ph" "ps") "ph" ("wn" "ps") ("wn" "ph") ("idd" ("wi" "ph" "ps") ("wn" "ph")) ("con3" "ph" "ps"))) ;; Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) (theorem "pm2_65i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm2_65i_1" ("wi" "ph" "ps")) ("pm2_65i_2" ("wi" "ph" ("wn" "ps")))) (for) ("wn" "ph") ("pm2_61i" "ps" ("wn" "ph") ("con2i" "ph" "ps" "pm2_65i_2") ("con3i" "ph" "ps" "pm2_65i_1"))) ;; A contradiction implies anything. Deduction from ~ pm2.21 . (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 22-Jul-2019.) (theorem "pm2_21dd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_21dd_1" ("wi" "ph" "ps")) ("pm2_21dd_2" ("wi" "ph" ("wn" "ps")))) (for) ("wi" "ph" "ch") ("pm2_21i" "ph" "ch" ("pm2_65i" "ph" "ps" "pm2_21dd_1" "pm2_21dd_2"))) ;; Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.) (theorem "pm2_65d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_65d_1" ("wi" "ph" ("wi" "ps" "ch"))) ("pm2_65d_2" ("wi" "ph" ("wi" "ps" ("wn" "ch"))))) (for) ("wi" "ph" ("wn" "ps")) ("pm2_01d" "ph" "ps" ("nsyld" "ph" "ps" "ch" "ps" "pm2_65d_2" "pm2_65d_1"))) ;; The rule of modus tollens. The rule says, "if ` ps ` is not true, and ` ph ` implies ` ps ` , then ` ph ` must also be not true." Modus tollens is short for "modus tollendo tollens," a Latin phrase that means "the mode that by denying denies" - remark in [Sanford] p. 39. It is also called denying the consequent. Modus tollens is closely related to modus ponens ~ ax-mp . Note that this rule is also valid in intuitionistic logic. Inference associated with ~ con3i . (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) (theorem "mto" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("mto_1" ("wn" "ps")) ("mto_2" ("wi" "ph" "ps"))) (for) ("wn" "ph") ("pm2_65i" "ph" "ps" "mto_2" ("a1i" ("wn" "ps") "ph" "mto_1"))) ;; Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) (theorem "mtod" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mtod_1" ("wi" "ph" ("wn" "ch"))) ("mtod_2" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wn" "ps")) ("pm2_65d" "ph" "ps" "ch" "mtod_2" ("a1d" "ph" ("wn" "ch") "ps" "mtod_1"))) ;; Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.) (theorem "mtoi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mtoi_1" ("wn" "ch")) ("mtoi_2" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wn" "ps")) ("mtod" "ph" "ps" "ch" ("a1i" ("wn" "ch") "ph" "mtoi_1") "mtoi_2")) ;; A rule similar to modus tollens. Inference associated with ~ con2i . (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.) (theorem "mt2" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("mt2_1" "ps") ("mt2_2" ("wi" "ph" ("wn" "ps")))) (for) ("wn" "ph") ("pm2_65i" "ph" "ps" ("a1i" "ps" "ph" "mt2_1") "mt2_2")) ;; A rule similar to modus tollens. Inference associated with ~ con1i . (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) (theorem "mt3" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("mt3_1" ("wn" "ps")) ("mt3_2" ("wi" ("wn" "ph") "ps"))) (for) "ph" ("notnotri" "ph" ("mto" ("wn" "ph") "ps" "mt3_1" "mt3_2"))) ;; Peirce's axiom. This odd-looking theorem is the "difference" between an intuitionistic system of propositional calculus and a classical system and is not accepted by intuitionists. When Peirce's axiom is added to an intuitionistic system, the system becomes equivalent to our classical system ~ ax-1 through ~ ax-3 . A notable fact about this theorem is that it requires ~ ax-3 for its proof even though the result has no negation connectives in it. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 9-Oct-2012.) (theorem "peirce" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" ("wi" "ph" "ps") "ph") "ph") ("ja" ("wi" "ph" "ps") "ph" "ph" ("simplim" "ph" "ps") ("id" "ph"))) ;; The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using ~ dfor2 , we can see that this essentially expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108. It is a special instance of the axiom "Roll", see ~ peirceroll . (Contributed by NM, 12-Aug-2004.) (theorem "looinv" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" ("wi" "ph" "ps") "ps") ("wi" ("wi" "ps" "ph") "ph")) ("syl6" ("wi" ("wi" "ph" "ps") "ps") ("wi" "ps" "ph") ("wi" ("wi" "ph" "ps") "ph") "ph" ("imim1" ("wi" "ph" "ps") "ps" "ph") ("peirce" "ph" "ps"))) ;; Theorem used to justify definition of biconditional ~ df-bi . (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (theorem "bijust" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wn" ("wi" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))))) ("mt2" ("wi" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))) ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("id" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("pm2_01" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))))) ;; Extend our wff definition to include the biconditional connective. (term "wb" ( ( "wff") ( "wff") ( "wff"))) ;; Define the biconditional (logical 'iff'). The definition ~ df-bi in this section is our first definition, which introduces and defines the biconditional connective ` <-> ` . We define a wff of the form ` ( ph <-> ps ) ` as an abbreviation for ` -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ` . Unlike most traditional developments, we have chosen not to have a separate symbol such as "Df." to mean "is defined as." Instead, we will later use the biconditional connective for this purpose ( ~ df-or is its first use), as it allows us to use logic to manipulate definitions directly. This greatly simplifies many proofs since it eliminates the need for a separate mechanism for introducing and eliminating definitions. Of course, we cannot use this mechanism to define the biconditional itself, since it hasn't been introduced yet. Instead, we use a more general form of definition, described as follows. In its most general form, a definition is simply an assertion that introduces a new symbol (or a new combination of existing symbols, as in ~ df-3an ) that is eliminable and does not strengthen the existing language. The latter requirement means that the set of provable statements not containing the new symbol (or new combination) should remain exactly the same after the definition is introduced. Our definition of the biconditional may look unusual compared to most definitions, but it strictly satisfies these requirements. The justification for our definition is that if we mechanically replace ` ( ph <-> ps ) ` (the definiendum i.e. the thing being defined) with ` -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ` (the definiens i.e. the defining expression) in the definition, the definition becomes the previously proved theorem ~ bijust . It is impossible to use ~ df-bi to prove any statement expressed in the original language that can't be proved from the original axioms, because if we simply replace each instance of ~ df-bi in the proof with the corresponding ~ bijust instance, we will end up with a proof from the original axioms. Note that from Metamath's point of view, a definition is just another axiom - i.e. an assertion we claim to be true - but from our high level point of view, we are not strengthening the language. To indicate this fact, we prefix definition labels with "df-" instead of "ax-". (This prefixing is an informal convention that means nothing to the Metamath proof verifier; it is just a naming convention for human readability.) After we define the constant true ` T. ` ( ~ df-tru ) and the constant false ` F. ` ( ~ df-fal ), we will be able to prove these truth table values: ` ( ( T. <-> T. ) <-> T. ) ` ( ~ trubitru ), ` ( ( T. <-> F. ) <-> F. ) ` ( ~ trubifal ), ` ( ( F. <-> T. ) <-> F. ) ` ( ~ falbitru ), and ` ( ( F. <-> F. ) <-> T. ) ` ( ~ falbifal ). See ~ dfbi1 , ~ dfbi2 , and ~ dfbi3 for theorems suggesting typical textbook definitions of ` <-> ` , showing that our definition has the properties we expect. Theorem ~ dfbi1 is particularly useful if we want to eliminate ` <-> ` from an expression to convert it to primitives. Theorem ~ dfbi shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding. Contrast with ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), ` -/\ ` ( ~ df-nan ), and ` \/_ ` ( ~ df-xor ) . In some sense ` <-> ` returns true if two truth values are equal; ` = ` ( ~ df-cleq ) returns true if two classes are equal. (Contributed by NM, 27-Dec-1992.) (axiom "df_bi" (!! ( ("ph" ( "wff")) ("ps" ( "wff"))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))))) ;; Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (theorem "impbi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wi" "ps" "ph") ("wb" "ph" "ps"))) ("expi" ("wi" "ph" "ps") ("wi" "ps" "ph") ("wb" "ph" "ps") ("ax_mp" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")) ("df_bi" "ph" "ps") ("simprim" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ;; Infer an equivalence from an implication and its converse. Inference associated with ~ impbi . (Contributed by NM, 29-Dec-1992.) (theorem "impbii" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("impbii_1" ("wi" "ph" "ps")) ("impbii_2" ("wi" "ps" "ph"))) (for) ("wb" "ph" "ps") ("mp2" ("wi" "ph" "ps") ("wi" "ps" "ph") ("wb" "ph" "ps") "impbii_1" "impbii_2" ("impbi" "ph" "ps"))) ;; Deduce an equivalence from two implications. Double deduction associated with ~ impbi and ~ impbii . Deduction associated with ~ impbid . (Contributed by Rodolfo Medina, 12-Oct-2010.) (theorem "impbidd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("impbidd_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th")))) ("impbidd_2" ("wi" "ph" ("wi" "ps" ("wi" "th" "ch"))))) (for) ("wi" "ph" ("wi" "ps" ("wb" "ch" "th"))) ("syl6c" "ph" "ps" ("wi" "ch" "th") ("wi" "th" "ch") ("wb" "ch" "th") "impbidd_1" "impbidd_2" ("impbi" "ch" "th"))) ;; Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.) (theorem "impbid21d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("impbid21d_1" ("wi" "ps" ("wi" "ch" "th"))) ("impbid21d_2" ("wi" "ph" ("wi" "th" "ch")))) (for) ("wi" "ph" ("wi" "ps" ("wb" "ch" "th"))) ("impbidd" "ph" "ps" "ch" "th" ("a1i" ("wi" "ps" ("wi" "ch" "th")) "ph" "impbid21d_1") ("a1d" "ph" ("wi" "th" "ch") "ps" "impbid21d_2"))) ;; Deduce an equivalence from two implications. Deduction associated with ~ impbi and ~ impbii . (Contributed by NM, 24-Jan-1993.) Revised to prove it from ~ impbid21d . (Revised by Wolf Lammen, 3-Nov-2012.) (theorem "impbid" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("impbid_1" ("wi" "ph" ("wi" "ps" "ch"))) ("impbid_2" ("wi" "ph" ("wi" "ch" "ps")))) (for) ("wi" "ph" ("wb" "ps" "ch")) ("pm2_43i" "ph" ("wb" "ps" "ch") ("impbid21d" "ph" "ph" "ps" "ch" "impbid_1" "impbid_2"))) ;; Relate the biconditional connective to primitive connectives. See ~ dfbi1ALT for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992.) (theorem "dfbi1" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("impbii" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("ax_mp" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("df_bi" "ph" "ps") ("simplim" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("impi" ("wi" "ph" "ps") ("wi" "ps" "ph") ("wb" "ph" "ps") ("impbi" "ph" "ps")))) ;; Alternate proof of ~ dfbi1 . This proof, discovered by Gregory Bush on 8-Mar-2004, has several curious properties. First, it has only 17 steps directly from the axioms and ~ df-bi , compared to over 800 steps were the proof of ~ dfbi1 expanded into axioms. Second, step 2 demands only the property of "true"; any axiom (or theorem) could be used. It might be thought, therefore, that it is in some sense redundant, but in fact no proof is shorter than this (measured by number of steps). Third, it illustrates how intermediate steps can "blow up" in size even in short proofs. Fourth, the compressed proof is only 182 bytes (or 17 bytes in D-proof notation), but the generated web page is over 200kB with intermediate steps that are essentially incomprehensible to humans (other than Gregory Bush). If there were an obfuscated code contest for proofs, this would be a contender. This "blowing up" and incomprehensibility of the intermediate steps vividly demonstrate the advantages of using many layered intermediate theorems, since each theorem is easier to understand. (Contributed by Gregory Bush, 10-Mar-2004.) (New usage is discouraged.) (Proof modification is discouraged.) (theorem "dfbi1ALT" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("ax_mp" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("df_bi" "ph" "ps") (forget (fn ("ch" "wff") ("th" "wff") ("ax_mp" ("wi" "ch" ("wi" "th" "ch")) ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))) ("ax_1" "ch" "th") ("ax_mp" ("wi" ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))) ("wn" ("wi" "ch" ("wi" "th" "ch")))) ("wi" ("wi" "ch" ("wi" "th" "ch")) ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))) ("ax_mp" ("wi" ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))) ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))))) ("wi" ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))) ("wn" ("wi" "ch" ("wi" "th" "ch")))) ("ax_1" ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))) ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))))) ("ax_mp" ("wi" ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))) ("wi" ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))))) ("wn" ("wi" "ch" ("wi" "th" "ch"))))) ("wi" ("wi" ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))) ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))))) ("wi" ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))) ("wn" ("wi" "ch" ("wi" "th" "ch"))))) ("ax_mp" ("wi" ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))))) ("wn" ("wi" "ch" ("wi" "th" "ch")))) ("wi" ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))) ("wi" ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))))) ("wn" ("wi" "ch" ("wi" "th" "ch"))))) ("ax_mp" ("wi" ("wn" ("wn" ("wi" "ch" ("wi" "th" "ch")))) ("wn" ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))))))) ("wi" ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))))) ("wn" ("wi" "ch" ("wi" "th" "ch")))) ("ax_mp" ("wn" ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))))) ("wi" ("wn" ("wn" ("wi" "ch" ("wi" "th" "ch")))) ("wn" ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))))))) ("df_bi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("ax_1" ("wn" ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))))) ("wn" ("wn" ("wi" "ch" ("wi" "th" "ch")))))) ("ax_3" ("wn" ("wi" "ch" ("wi" "th" "ch"))) ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))))))) ("ax_1" ("wi" ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))))) ("wn" ("wi" "ch" ("wi" "th" "ch")))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))))) ("ax_2" ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))))) ("wi" ("wi" ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps")))))) ("wn" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))))) ("wn" ("wi" "ch" ("wi" "th" "ch")))))) ("ax_3" ("wi" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wb" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))))) ("wi" "ch" ("wi" "th" "ch"))))))))) ;; Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (theorem "biimp" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wi" "ph" "ps")) ("syl" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wi" "ph" "ps") ("ax_mp" ("wn" ("wi" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("df_bi" "ph" "ps") ("simplim" ("wi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ("wn" ("wi" ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wb" "ph" "ps"))))) ("simplim" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph"))))) ;; Infer an implication from a logical equivalence. Inference associated with ~ biimp . (Contributed by NM, 29-Dec-1992.) (theorem "biimpi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("biimpi_1" ("wb" "ph" "ps"))) (for) ("wi" "ph" "ps") ("ax_mp" ("wb" "ph" "ps") ("wi" "ph" "ps") "biimpi_1" ("biimp" "ph" "ps"))) ;; A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 3-Jan-1993.) (theorem "sylbi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylbi_1" ("wb" "ph" "ps")) ("sylbi_2" ("wi" "ps" "ch"))) (for) ("wi" "ph" "ch") ("syl" "ph" "ps" "ch" ("biimpi" "ph" "ps" "sylbi_1") "sylbi_2")) ;; A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993.) (theorem "sylib" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylib_1" ("wi" "ph" "ps")) ("sylib_2" ("wb" "ps" "ch"))) (for) ("wi" "ph" "ch") ("syl" "ph" "ps" "ch" "sylib_1" ("biimpi" "ps" "ch" "sylib_2"))) ;; A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019.) (theorem "sylbb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylbb_1" ("wb" "ph" "ps")) ("sylbb_2" ("wb" "ps" "ch"))) (for) ("wi" "ph" "ch") ("sylbi" "ph" "ps" "ch" "sylbb_1" ("biimpi" "ps" "ch" "sylbb_2"))) ;; Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) (theorem "biimpr" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wi" "ps" "ph")) ("sylbi" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wi" "ps" "ph") ("dfbi1" "ph" "ps") ("simprim" ("wi" "ph" "ps") ("wi" "ps" "ph")))) ;; Commutative law for the biconditional. (Contributed by Wolf Lammen, 10-Nov-2012.) (theorem "bicom1" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wb" "ps" "ph")) ("impbid" ("wb" "ph" "ps") "ps" "ph" ("biimpr" "ph" "ps") ("biimp" "ph" "ps"))) ;; Commutative law for the biconditional. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.) (theorem "bicom" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" "ph" "ps") ("wb" "ps" "ph")) ("impbii" ("wb" "ph" "ps") ("wb" "ps" "ph") ("bicom1" "ph" "ps") ("bicom1" "ps" "ph"))) ;; Commute two sides of a biconditional in a deduction. (Contributed by NM, 14-May-1993.) (theorem "bicomd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("bicomd_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" "ch" "ps")) ("sylib" "ph" ("wb" "ps" "ch") ("wb" "ch" "ps") "bicomd_1" ("bicom" "ps" "ch"))) ;; Inference from commutative law for logical equivalence. (Contributed by NM, 3-Jan-1993.) (theorem "bicomi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("bicomi_1" ("wb" "ph" "ps"))) (for) ("wb" "ps" "ph") ("ax_mp" ("wb" "ph" "ps") ("wb" "ps" "ph") "bicomi_1" ("bicom1" "ph" "ps"))) ;; Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (theorem "impbid1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("impbid1_1" ("wi" "ph" ("wi" "ps" "ch"))) ("impbid1_2" ("wi" "ch" "ps"))) (for) ("wi" "ph" ("wb" "ps" "ch")) ("impbid" "ph" "ps" "ch" "impbid1_1" ("a1i" ("wi" "ch" "ps") "ph" "impbid1_2"))) ;; Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) (theorem "impbid2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("impbid2_1" ("wi" "ps" "ch")) ("impbid2_2" ("wi" "ph" ("wi" "ch" "ps")))) (for) ("wi" "ph" ("wb" "ps" "ch")) ("bicomd" "ph" "ch" "ps" ("impbid1" "ph" "ch" "ps" "impbid2_2" "impbid2_1"))) ;; A variation on ~ impbid with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.) (theorem "impcon4bid" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("impcon4bid_1" ("wi" "ph" ("wi" "ps" "ch"))) ("impcon4bid_2" ("wi" "ph" ("wi" ("wn" "ps") ("wn" "ch"))))) (for) ("wi" "ph" ("wb" "ps" "ch")) ("impbid" "ph" "ps" "ch" "impcon4bid_1" ("con4d" "ph" "ps" "ch" "impcon4bid_2"))) ;; Infer a converse implication from a logical equivalence. Inference associated with ~ biimpr . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 16-Sep-2013.) (theorem "biimpri" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("biimpri_1" ("wb" "ph" "ps"))) (for) ("wi" "ps" "ph") ("biimpi" "ps" "ph" ("bicomi" "ph" "ps" "biimpri_1"))) ;; Deduce an implication from a logical equivalence. Deduction associated with ~ biimp and ~ biimpi . (Contributed by NM, 11-Jan-1993.) (theorem "biimpd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("biimpd_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wi" "ps" "ch")) ("syl" "ph" ("wb" "ps" "ch") ("wi" "ps" "ch") "biimpd_1" ("biimp" "ps" "ch"))) ;; An inference from a biconditional, related to modus ponens. (Contributed by NM, 11-May-1993.) (theorem "mpbi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("mpbi_min" "ph") ("mpbi_maj" ("wb" "ph" "ps"))) (for) "ps" ("ax_mp" "ph" "ps" "mpbi_min" ("biimpi" "ph" "ps" "mpbi_maj"))) ;; An inference from a biconditional, related to modus ponens. (Contributed by NM, 28-Dec-1992.) (theorem "mpbir" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("mpbir_min" "ps") ("mpbir_maj" ("wb" "ph" "ps"))) (for) "ph" ("ax_mp" "ps" "ph" "mpbir_min" ("biimpri" "ph" "ps" "mpbir_maj"))) ;; A deduction from a biconditional, related to modus ponens. (Contributed by NM, 21-Jun-1993.) (theorem "mpbid" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpbid_min" ("wi" "ph" "ps")) ("mpbid_maj" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" "ch") ("mpd" "ph" "ps" "ch" "mpbid_min" ("biimpd" "ph" "ps" "ch" "mpbid_maj"))) ;; An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) (theorem "mpbii" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpbii_min" "ps") ("mpbii_maj" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" "ch") ("mpbid" "ph" "ps" "ch" ("a1i" "ps" "ph" "mpbii_min") "mpbii_maj")) ;; A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 3-Jan-1993.) (theorem "sylibr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylibr_1" ("wi" "ph" "ps")) ("sylibr_2" ("wb" "ch" "ps"))) (for) ("wi" "ph" "ch") ("syl" "ph" "ps" "ch" "sylibr_1" ("biimpri" "ch" "ps" "sylibr_2"))) ;; A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 3-Jan-1993.) (theorem "sylbir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylbir_1" ("wb" "ps" "ph")) ("sylbir_2" ("wi" "ps" "ch"))) (for) ("wi" "ph" "ch") ("syl" "ph" "ps" "ch" ("biimpri" "ps" "ph" "sylbir_1") "sylbir_2")) ;; A mixed syllogism inference from two biconditionals. Note on the various syllogism-like statements in set.mm. The hypothetical syllogism ~ syl infers an implication from two implications (and there are ~ 3syl and ~ 4syl for chaining more inferences). There are four inferences inferring an implication from one implication and one biconditional: ~ sylbi , ~ sylib , ~ sylbir , ~ sylibr ; four inferences inferring an implication from two biconditionals: ~ sylbb , ~ sylbbr , ~ sylbb1 , ~ sylbb2 ; four inferences inferring a biconditional from two biconditionals: ~ bitri , ~ bitr2i , ~ bitr3i , ~ bitr4i (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, ~ syld , ~ syl5 , ~ syl6 , ~ mpbid , ~ bitrd , ~ syl5bb , ~ syl6bb and variants. (Contributed by BJ, 21-Apr-2019.) (theorem "sylbbr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylbbr_1" ("wb" "ph" "ps")) ("sylbbr_2" ("wb" "ps" "ch"))) (for) ("wi" "ch" "ph") ("sylibr" "ch" "ps" "ph" ("biimpri" "ps" "ch" "sylbbr_2") "sylbbr_1")) ;; A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.) (theorem "sylbb1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylbb1_1" ("wb" "ph" "ps")) ("sylbb1_2" ("wb" "ph" "ch"))) (for) ("wi" "ps" "ch") ("sylib" "ps" "ph" "ch" ("biimpri" "ph" "ps" "sylbb1_1") "sylbb1_2")) ;; A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.) (theorem "sylbb2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylbb2_1" ("wb" "ph" "ps")) ("sylbb2_2" ("wb" "ch" "ps"))) (for) ("wi" "ph" "ch") ("sylbi" "ph" "ps" "ch" "sylbb2_1" ("biimpri" "ch" "ps" "sylbb2_2"))) ;; A syllogism deduction. (Contributed by NM, 3-Aug-1994.) (theorem "sylibd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylibd_1" ("wi" "ph" ("wi" "ps" "ch"))) ("sylibd_2" ("wi" "ph" ("wb" "ch" "th")))) (for) ("wi" "ph" ("wi" "ps" "th")) ("syld" "ph" "ps" "ch" "th" "sylibd_1" ("biimpd" "ph" "ch" "th" "sylibd_2"))) ;; A syllogism deduction. (Contributed by NM, 3-Aug-1994.) (theorem "sylbid" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylbid_1" ("wi" "ph" ("wb" "ps" "ch"))) ("sylbid_2" ("wi" "ph" ("wi" "ch" "th")))) (for) ("wi" "ph" ("wi" "ps" "th")) ("syld" "ph" "ps" "ch" "th" ("biimpd" "ph" "ps" "ch" "sylbid_1") "sylbid_2")) ;; A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.) (theorem "mpbidi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpbidi_min" ("wi" "th" ("wi" "ph" "ps"))) ("mpbidi_maj" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "th" ("wi" "ph" "ch")) ("sylcom" "th" "ph" "ps" "ch" "mpbidi_min" ("biimpd" "ph" "ps" "ch" "mpbidi_maj"))) ;; A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 12-Jan-1993.) (theorem "syl5bi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5bi_1" ("wb" "ph" "ps")) ("syl5bi_2" ("wi" "ch" ("wi" "ps" "th")))) (for) ("wi" "ch" ("wi" "ph" "th")) ("syl5" "ph" "ps" "ch" "th" ("biimpi" "ph" "ps" "syl5bi_1") "syl5bi_2")) ;; A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.) (theorem "syl5bir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5bir_1" ("wb" "ps" "ph")) ("syl5bir_2" ("wi" "ch" ("wi" "ps" "th")))) (for) ("wi" "ch" ("wi" "ph" "th")) ("syl5" "ph" "ps" "ch" "th" ("biimpri" "ps" "ph" "syl5bir_1") "syl5bir_2")) ;; A mixed syllogism inference. (Contributed by NM, 12-Jan-1993.) (theorem "syl5ib" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5ib_1" ("wi" "ph" "ps")) ("syl5ib_2" ("wi" "ch" ("wb" "ps" "th")))) (for) ("wi" "ch" ("wi" "ph" "th")) ("syl5" "ph" "ps" "ch" "th" "syl5ib_1" ("biimpd" "ch" "ps" "th" "syl5ib_2"))) ;; A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.) (theorem "syl5ibcom" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5ib_1" ("wi" "ph" "ps")) ("syl5ib_2" ("wi" "ch" ("wb" "ps" "th")))) (for) ("wi" "ph" ("wi" "ch" "th")) ("com12" "ch" "ph" "th" ("syl5ib" "ph" "ps" "ch" "th" "syl5ib_1" "syl5ib_2"))) ;; A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) (theorem "syl5ibr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5ibr_1" ("wi" "ph" "th")) ("syl5ibr_2" ("wi" "ch" ("wb" "ps" "th")))) (for) ("wi" "ch" ("wi" "ph" "ps")) ("syl5ib" "ph" "th" "ch" "ps" "syl5ibr_1" ("bicomd" "ch" "ps" "th" "syl5ibr_2"))) ;; A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.) (theorem "syl5ibrcom" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5ibr_1" ("wi" "ph" "th")) ("syl5ibr_2" ("wi" "ch" ("wb" "ps" "th")))) (for) ("wi" "ph" ("wi" "ch" "ps")) ("com12" "ch" "ph" "ps" ("syl5ibr" "ph" "ps" "ch" "th" "syl5ibr_1" "syl5ibr_2"))) ;; Deduce a converse implication from a logical equivalence. Deduction associated with ~ biimpr and ~ biimpri . (Contributed by NM, 11-Jan-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) (theorem "biimprd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("biimprd_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wi" "ch" "ps")) ("syl5ibr" "ch" "ps" "ph" "ch" ("id" "ch") "biimprd_1")) ;; Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) (theorem "biimpcd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("biimpcd_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ps" ("wi" "ph" "ch")) ("syl5ibcom" "ps" "ps" "ph" "ch" ("id" "ps") "biimpcd_1")) ;; Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) (theorem "biimprcd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("biimpcd_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ch" ("wi" "ph" "ps")) ("syl5ibrcom" "ch" "ps" "ph" "ch" ("id" "ch") "biimpcd_1")) ;; A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.) (theorem "syl6ib" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl6ib_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl6ib_2" ("wb" "ch" "th"))) (for) ("wi" "ph" ("wi" "ps" "th")) ("syl6" "ph" "ps" "ch" "th" "syl6ib_1" ("biimpi" "ch" "th" "syl6ib_2"))) ;; A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. (Contributed by NM, 10-Jan-1993.) (theorem "syl6ibr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl6ibr_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl6ibr_2" ("wb" "th" "ch"))) (for) ("wi" "ph" ("wi" "ps" "th")) ("syl6" "ph" "ps" "ch" "th" "syl6ibr_1" ("biimpri" "th" "ch" "syl6ibr_2"))) ;; A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.) (theorem "syl6bi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl6bi_1" ("wi" "ph" ("wb" "ps" "ch"))) ("syl6bi_2" ("wi" "ch" "th"))) (for) ("wi" "ph" ("wi" "ps" "th")) ("syl6" "ph" "ps" "ch" "th" ("biimpd" "ph" "ps" "ch" "syl6bi_1") "syl6bi_2")) ;; A mixed syllogism inference. (Contributed by NM, 18-May-1994.) (theorem "syl6bir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl6bir_1" ("wi" "ph" ("wb" "ch" "ps"))) ("syl6bir_2" ("wi" "ch" "th"))) (for) ("wi" "ph" ("wi" "ps" "th")) ("syl6" "ph" "ps" "ch" "th" ("biimprd" "ph" "ch" "ps" "syl6bir_1") "syl6bir_2")) ;; A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-1993.) (theorem "syl7bi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl7bi_1" ("wb" "ph" "ps")) ("syl7bi_2" ("wi" "ch" ("wi" "th" ("wi" "ps" "ta"))))) (for) ("wi" "ch" ("wi" "th" ("wi" "ph" "ta"))) ("syl7" "ph" "ps" "ch" "th" "ta" ("biimpi" "ph" "ps" "syl7bi_1") "syl7bi_2")) ;; A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (theorem "syl8ib" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl8ib_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th")))) ("syl8ib_2" ("wb" "th" "ta"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "ta"))) ("syl8" "ph" "ps" "ch" "th" "ta" "syl8ib_1" ("biimpi" "th" "ta" "syl8ib_2"))) ;; A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (theorem "mpbird" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpbird_min" ("wi" "ph" "ch")) ("mpbird_maj" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" "ps") ("mpd" "ph" "ch" "ps" "mpbird_min" ("biimprd" "ph" "ps" "ch" "mpbird_maj"))) ;; An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) (theorem "mpbiri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpbiri_min" "ch") ("mpbiri_maj" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" "ps") ("mpbird" "ph" "ps" "ch" ("a1i" "ch" "ph" "mpbiri_min") "mpbiri_maj")) ;; A syllogism deduction. (Contributed by NM, 3-Aug-1994.) (theorem "sylibrd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylibrd_1" ("wi" "ph" ("wi" "ps" "ch"))) ("sylibrd_2" ("wi" "ph" ("wb" "th" "ch")))) (for) ("wi" "ph" ("wi" "ps" "th")) ("syld" "ph" "ps" "ch" "th" "sylibrd_1" ("biimprd" "ph" "th" "ch" "sylibrd_2"))) ;; A syllogism deduction. (Contributed by NM, 3-Aug-1994.) (theorem "sylbird" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylbird_1" ("wi" "ph" ("wb" "ch" "ps"))) ("sylbird_2" ("wi" "ph" ("wi" "ch" "th")))) (for) ("wi" "ph" ("wi" "ps" "th")) ("syld" "ph" "ps" "ch" "th" ("biimprd" "ph" "ch" "ps" "sylbird_1") "sylbird_2")) ;; Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. This is part of Frege's eighth axiom per Proposition 54 of [Frege1879] p. 50; see also ~ eqid . (Contributed by NM, 2-Jun-1993.) (theorem "biid" (for ("ph" ( "wff"))) (for) (for) ("wb" "ph" "ph") ("impbii" "ph" "ph" ("id" "ph") ("id" "ph"))) ;; Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.) (theorem "biidd" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wb" "ps" "ps")) ("a1i" ("wb" "ps" "ps") "ph" ("biid" "ps"))) ;; Two propositions are equivalent if they are both true. Closed form of ~ 2th . Equivalent to a ~ biimp -like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version ` ( ph <-> ( ps <-> ( ph <-> ps ) ) ) ` . (Contributed by Wolf Lammen, 12-May-2013.) (theorem "pm5_1im" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wi" "ps" ("wb" "ph" "ps"))) ("impbid21d" "ph" "ps" "ph" "ps" ("ax_1" "ps" "ph") ("ax_1" "ph" "ps"))) ;; Two truths are equivalent. (Contributed by NM, 18-Aug-1993.) (theorem "_2th" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("_2th_1" "ph") ("_2th_2" "ps")) (for) ("wb" "ph" "ps") ("impbii" "ph" "ps" ("a1i" "ps" "ph" "_2th_2") ("a1i" "ph" "ps" "_2th_1"))) ;; Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.) (theorem "_2thd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_2thd_1" ("wi" "ph" "ps")) ("_2thd_2" ("wi" "ph" "ch"))) (for) ("wi" "ph" ("wb" "ps" "ch")) ("sylc" "ph" "ps" "ch" ("wb" "ps" "ch") "_2thd_1" "_2thd_2" ("pm5_1im" "ps" "ch"))) ;; Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.) (theorem "ibi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("ibi_1" ("wi" "ph" ("wb" "ph" "ps")))) (for) ("wi" "ph" "ps") ("pm2_43i" "ph" "ps" ("biimpd" "ph" "ph" "ps" "ibi_1"))) ;; Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.) (theorem "ibir" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("ibir_1" ("wi" "ph" ("wb" "ps" "ph")))) (for) ("wi" "ph" "ps") ("ibi" "ph" "ps" ("bicomd" "ph" "ps" "ph" "ibir_1"))) ;; Deduction that converts a biconditional implied by one of its arguments, into an implication. Deduction associated with ~ ibi . (Contributed by NM, 26-Jun-2004.) (theorem "ibd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ibd_1" ("wi" "ph" ("wi" "ps" ("wb" "ps" "ch"))))) (for) ("wi" "ph" ("wi" "ps" "ch")) ("syli" "ps" "ph" ("wb" "ps" "ch") "ch" "ibd_1" ("biimp" "ps" "ch"))) ;; Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.) (theorem "pm5_74" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wb" "ps" "ch")) ("wb" ("wi" "ph" "ps") ("wi" "ph" "ch"))) ("impbii" ("wi" "ph" ("wb" "ps" "ch")) ("wb" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("impbid" ("wi" "ph" ("wb" "ps" "ch")) ("wi" "ph" "ps") ("wi" "ph" "ch") ("imim3i" ("wb" "ps" "ch") "ps" "ch" "ph" ("biimp" "ps" "ch")) ("imim3i" ("wb" "ps" "ch") "ch" "ps" "ph" ("biimpr" "ps" "ch"))) ("impbidd" ("wb" ("wi" "ph" "ps") ("wi" "ph" "ch")) "ph" "ps" "ch" ("pm2_86d" ("wb" ("wi" "ph" "ps") ("wi" "ph" "ch")) "ph" "ps" "ch" ("biimp" ("wi" "ph" "ps") ("wi" "ph" "ch"))) ("pm2_86d" ("wb" ("wi" "ph" "ps") ("wi" "ph" "ch")) "ph" "ch" "ps" ("biimpr" ("wi" "ph" "ps") ("wi" "ph" "ch")))))) ;; Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.) (theorem "pm5_74i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm5_74i_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wb" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("mpbi" ("wi" "ph" ("wb" "ps" "ch")) ("wb" ("wi" "ph" "ps") ("wi" "ph" "ch")) "pm5_74i_1" ("pm5_74" "ph" "ps" "ch"))) ;; Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.) (theorem "pm5_74ri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm5_74ri_1" ("wb" ("wi" "ph" "ps") ("wi" "ph" "ch")))) (for) ("wi" "ph" ("wb" "ps" "ch")) ("mpbir" ("wi" "ph" ("wb" "ps" "ch")) ("wb" ("wi" "ph" "ps") ("wi" "ph" "ch")) "pm5_74ri_1" ("pm5_74" "ph" "ps" "ch"))) ;; Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.) (theorem "pm5_74d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm5_74d_1" ("wi" "ph" ("wi" "ps" ("wb" "ch" "th"))))) (for) ("wi" "ph" ("wb" ("wi" "ps" "ch") ("wi" "ps" "th"))) ("sylib" "ph" ("wi" "ps" ("wb" "ch" "th")) ("wb" ("wi" "ps" "ch") ("wi" "ps" "th")) "pm5_74d_1" ("pm5_74" "ps" "ch" "th"))) ;; Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.) (theorem "pm5_74rd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm5_74rd_1" ("wi" "ph" ("wb" ("wi" "ps" "ch") ("wi" "ps" "th"))))) (for) ("wi" "ph" ("wi" "ps" ("wb" "ch" "th"))) ("sylibr" "ph" ("wb" ("wi" "ps" "ch") ("wi" "ps" "th")) ("wi" "ps" ("wb" "ch" "th")) "pm5_74rd_1" ("pm5_74" "ps" "ch" "th"))) ;; An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) (theorem "bitri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("bitri_1" ("wb" "ph" "ps")) ("bitri_2" ("wb" "ps" "ch"))) (for) ("wb" "ph" "ch") ("impbii" "ph" "ch" ("sylbb" "ph" "ps" "ch" "bitri_1" "bitri_2") ("sylbbr" "ph" "ps" "ch" "bitri_1" "bitri_2"))) ;; An inference from transitive law for logical equivalence. (Contributed by NM, 12-Mar-1993.) (theorem "bitr2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("bitr2i_1" ("wb" "ph" "ps")) ("bitr2i_2" ("wb" "ps" "ch"))) (for) ("wb" "ch" "ph") ("bicomi" "ph" "ch" ("bitri" "ph" "ps" "ch" "bitr2i_1" "bitr2i_2"))) ;; An inference from transitive law for logical equivalence. (Contributed by NM, 2-Jun-1993.) (theorem "bitr3i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("bitr3i_1" ("wb" "ps" "ph")) ("bitr3i_2" ("wb" "ps" "ch"))) (for) ("wb" "ph" "ch") ("bitri" "ph" "ps" "ch" ("bicomi" "ps" "ph" "bitr3i_1") "bitr3i_2")) ;; An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.) (theorem "bitr4i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("bitr4i_1" ("wb" "ph" "ps")) ("bitr4i_2" ("wb" "ch" "ps"))) (for) ("wb" "ph" "ch") ("bitri" "ph" "ps" "ch" "bitr4i_1" ("bicomi" "ch" "ps" "bitr4i_2"))) ;; Deduction form of ~ bitri . (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) (theorem "bitrd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("bitrd_1" ("wi" "ph" ("wb" "ps" "ch"))) ("bitrd_2" ("wi" "ph" ("wb" "ch" "th")))) (for) ("wi" "ph" ("wb" "ps" "th")) ("pm5_74ri" "ph" "ps" "th" ("bitri" ("wi" "ph" "ps") ("wi" "ph" "ch") ("wi" "ph" "th") ("pm5_74i" "ph" "ps" "ch" "bitrd_1") ("pm5_74i" "ph" "ch" "th" "bitrd_2")))) ;; Deduction form of ~ bitr2i . (Contributed by NM, 9-Jun-2004.) (theorem "bitr2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("bitr2d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("bitr2d_2" ("wi" "ph" ("wb" "ch" "th")))) (for) ("wi" "ph" ("wb" "th" "ps")) ("bicomd" "ph" "ps" "th" ("bitrd" "ph" "ps" "ch" "th" "bitr2d_1" "bitr2d_2"))) ;; Deduction form of ~ bitr3i . (Contributed by NM, 14-May-1993.) (theorem "bitr3d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("bitr3d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("bitr3d_2" ("wi" "ph" ("wb" "ps" "th")))) (for) ("wi" "ph" ("wb" "ch" "th")) ("bitrd" "ph" "ch" "ps" "th" ("bicomd" "ph" "ps" "ch" "bitr3d_1") "bitr3d_2")) ;; Deduction form of ~ bitr4i . (Contributed by NM, 30-Jun-1993.) (theorem "bitr4d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("bitr4d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("bitr4d_2" ("wi" "ph" ("wb" "th" "ch")))) (for) ("wi" "ph" ("wb" "ps" "th")) ("bitrd" "ph" "ps" "ch" "th" "bitr4d_1" ("bicomd" "ph" "th" "ch" "bitr4d_2"))) ;; A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.) (theorem "syl5bb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5bb_1" ("wb" "ph" "ps")) ("syl5bb_2" ("wi" "ch" ("wb" "ps" "th")))) (for) ("wi" "ch" ("wb" "ph" "th")) ("bitrd" "ch" "ph" "ps" "th" ("a1i" ("wb" "ph" "ps") "ch" "syl5bb_1") "syl5bb_2")) ;; A syllogism inference from two biconditionals. (Contributed by NM, 1-Aug-1993.) (theorem "syl5rbb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5rbb_1" ("wb" "ph" "ps")) ("syl5rbb_2" ("wi" "ch" ("wb" "ps" "th")))) (for) ("wi" "ch" ("wb" "th" "ph")) ("bicomd" "ch" "ph" "th" ("syl5bb" "ph" "ps" "ch" "th" "syl5rbb_1" "syl5rbb_2"))) ;; A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) (theorem "syl5bbr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5bbr_1" ("wb" "ps" "ph")) ("syl5bbr_2" ("wi" "ch" ("wb" "ps" "th")))) (for) ("wi" "ch" ("wb" "ph" "th")) ("syl5bb" "ph" "ps" "ch" "th" ("bicomi" "ps" "ph" "syl5bbr_1") "syl5bbr_2")) ;; A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) (theorem "syl5rbbr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl5rbbr_1" ("wb" "ps" "ph")) ("syl5rbbr_2" ("wi" "ch" ("wb" "ps" "th")))) (for) ("wi" "ch" ("wb" "th" "ph")) ("syl5rbb" "ph" "ps" "ch" "th" ("bicomi" "ps" "ph" "syl5rbbr_1") "syl5rbbr_2")) ;; A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.) (theorem "syl6bb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl6bb_1" ("wi" "ph" ("wb" "ps" "ch"))) ("syl6bb_2" ("wb" "ch" "th"))) (for) ("wi" "ph" ("wb" "ps" "th")) ("bitrd" "ph" "ps" "ch" "th" "syl6bb_1" ("a1i" ("wb" "ch" "th") "ph" "syl6bb_2"))) ;; A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) (theorem "syl6rbb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl6rbb_1" ("wi" "ph" ("wb" "ps" "ch"))) ("syl6rbb_2" ("wb" "ch" "th"))) (for) ("wi" "ph" ("wb" "th" "ps")) ("bicomd" "ph" "ps" "th" ("syl6bb" "ph" "ps" "ch" "th" "syl6rbb_1" "syl6rbb_2"))) ;; A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.) (theorem "syl6bbr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl6bbr_1" ("wi" "ph" ("wb" "ps" "ch"))) ("syl6bbr_2" ("wb" "th" "ch"))) (for) ("wi" "ph" ("wb" "ps" "th")) ("syl6bb" "ph" "ps" "ch" "th" "syl6bbr_1" ("bicomi" "th" "ch" "syl6bbr_2"))) ;; A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) (theorem "syl6rbbr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl6rbbr_1" ("wi" "ph" ("wb" "ps" "ch"))) ("syl6rbbr_2" ("wb" "th" "ch"))) (for) ("wi" "ph" ("wb" "th" "ps")) ("syl6rbb" "ph" "ps" "ch" "th" "syl6rbbr_1" ("bicomi" "th" "ch" "syl6rbbr_2"))) ;; A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.) (theorem "_3imtr3i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3imtr3_1" ("wi" "ph" "ps")) ("_3imtr3_2" ("wb" "ph" "ch")) ("_3imtr3_3" ("wb" "ps" "th"))) (for) ("wi" "ch" "th") ("sylib" "ch" "ps" "th" ("sylbir" "ch" "ph" "ps" "_3imtr3_2" "_3imtr3_1") "_3imtr3_3")) ;; A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 3-Jan-1993.) (theorem "_3imtr4i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3imtr4_1" ("wi" "ph" "ps")) ("_3imtr4_2" ("wb" "ch" "ph")) ("_3imtr4_3" ("wb" "th" "ps"))) (for) ("wi" "ch" "th") ("sylibr" "ch" "ps" "th" ("sylbi" "ch" "ph" "ps" "_3imtr4_2" "_3imtr4_1") "_3imtr4_3")) ;; More general version of ~ 3imtr3i . Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.) (theorem "_3imtr3d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3imtr3d_1" ("wi" "ph" ("wi" "ps" "ch"))) ("_3imtr3d_2" ("wi" "ph" ("wb" "ps" "th"))) ("_3imtr3d_3" ("wi" "ph" ("wb" "ch" "ta")))) (for) ("wi" "ph" ("wi" "th" "ta")) ("sylbird" "ph" "th" "ps" "ta" "_3imtr3d_2" ("sylibd" "ph" "ps" "ch" "ta" "_3imtr3d_1" "_3imtr3d_3"))) ;; More general version of ~ 3imtr4i . Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.) (theorem "_3imtr4d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3imtr4d_1" ("wi" "ph" ("wi" "ps" "ch"))) ("_3imtr4d_2" ("wi" "ph" ("wb" "th" "ps"))) ("_3imtr4d_3" ("wi" "ph" ("wb" "ta" "ch")))) (for) ("wi" "ph" ("wi" "th" "ta")) ("sylbid" "ph" "th" "ps" "ta" "_3imtr4d_2" ("sylibrd" "ph" "ps" "ch" "ta" "_3imtr4d_1" "_3imtr4d_3"))) ;; More general version of ~ 3imtr3i . Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) (theorem "_3imtr3g" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3imtr3g_1" ("wi" "ph" ("wi" "ps" "ch"))) ("_3imtr3g_2" ("wb" "ps" "th")) ("_3imtr3g_3" ("wb" "ch" "ta"))) (for) ("wi" "ph" ("wi" "th" "ta")) ("syl6ib" "ph" "th" "ch" "ta" ("syl5bir" "th" "ps" "ph" "ch" "_3imtr3g_2" "_3imtr3g_1") "_3imtr3g_3")) ;; More general version of ~ 3imtr4i . Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) (theorem "_3imtr4g" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3imtr4g_1" ("wi" "ph" ("wi" "ps" "ch"))) ("_3imtr4g_2" ("wb" "th" "ps")) ("_3imtr4g_3" ("wb" "ta" "ch"))) (for) ("wi" "ph" ("wi" "th" "ta")) ("syl6ibr" "ph" "th" "ch" "ta" ("syl5bi" "th" "ps" "ph" "ch" "_3imtr4g_2" "_3imtr4g_1") "_3imtr4g_3")) ;; A chained inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.) (theorem "_3bitri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3bitri_1" ("wb" "ph" "ps")) ("_3bitri_2" ("wb" "ps" "ch")) ("_3bitri_3" ("wb" "ch" "th"))) (for) ("wb" "ph" "th") ("bitri" "ph" "ps" "th" "_3bitri_1" ("bitri" "ps" "ch" "th" "_3bitri_2" "_3bitri_3"))) ;; A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) (theorem "_3bitrri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3bitri_1" ("wb" "ph" "ps")) ("_3bitri_2" ("wb" "ps" "ch")) ("_3bitri_3" ("wb" "ch" "th"))) (for) ("wb" "th" "ph") ("bitr3i" "th" "ch" "ph" "_3bitri_3" ("bitr2i" "ph" "ps" "ch" "_3bitri_1" "_3bitri_2"))) ;; A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) (theorem "_3bitr2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3bitr2i_1" ("wb" "ph" "ps")) ("_3bitr2i_2" ("wb" "ch" "ps")) ("_3bitr2i_3" ("wb" "ch" "th"))) (for) ("wb" "ph" "th") ("bitri" "ph" "ch" "th" ("bitr4i" "ph" "ps" "ch" "_3bitr2i_1" "_3bitr2i_2") "_3bitr2i_3")) ;; A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) (theorem "_3bitr2ri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3bitr2i_1" ("wb" "ph" "ps")) ("_3bitr2i_2" ("wb" "ch" "ps")) ("_3bitr2i_3" ("wb" "ch" "th"))) (for) ("wb" "th" "ph") ("bitr2i" "ph" "ch" "th" ("bitr4i" "ph" "ps" "ch" "_3bitr2i_1" "_3bitr2i_2") "_3bitr2i_3")) ;; A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.) (theorem "_3bitr3i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3bitr3i_1" ("wb" "ph" "ps")) ("_3bitr3i_2" ("wb" "ph" "ch")) ("_3bitr3i_3" ("wb" "ps" "th"))) (for) ("wb" "ch" "th") ("bitri" "ch" "ps" "th" ("bitr3i" "ch" "ph" "ps" "_3bitr3i_2" "_3bitr3i_1") "_3bitr3i_3")) ;; A chained inference from transitive law for logical equivalence. (Contributed by NM, 21-Jun-1993.) (theorem "_3bitr3ri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3bitr3i_1" ("wb" "ph" "ps")) ("_3bitr3i_2" ("wb" "ph" "ch")) ("_3bitr3i_3" ("wb" "ps" "th"))) (for) ("wb" "th" "ch") ("bitr3i" "th" "ps" "ch" "_3bitr3i_3" ("bitr3i" "ps" "ph" "ch" "_3bitr3i_1" "_3bitr3i_2"))) ;; A chained inference from transitive law for logical equivalence. This inference is frequently used to apply a definition to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (theorem "_3bitr4i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3bitr4i_1" ("wb" "ph" "ps")) ("_3bitr4i_2" ("wb" "ch" "ph")) ("_3bitr4i_3" ("wb" "th" "ps"))) (for) ("wb" "ch" "th") ("bitri" "ch" "ph" "th" "_3bitr4i_2" ("bitr4i" "ph" "ps" "th" "_3bitr4i_1" "_3bitr4i_3"))) ;; A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.) (theorem "_3bitr4ri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3bitr4i_1" ("wb" "ph" "ps")) ("_3bitr4i_2" ("wb" "ch" "ph")) ("_3bitr4i_3" ("wb" "th" "ps"))) (for) ("wb" "th" "ch") ("bitr2i" "ch" "ph" "th" "_3bitr4i_2" ("bitr4i" "ph" "ps" "th" "_3bitr4i_1" "_3bitr4i_3"))) ;; Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.) (theorem "_3bitrd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3bitrd_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3bitrd_2" ("wi" "ph" ("wb" "ch" "th"))) ("_3bitrd_3" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" "ps" "ta")) ("bitrd" "ph" "ps" "th" "ta" ("bitrd" "ph" "ps" "ch" "th" "_3bitrd_1" "_3bitrd_2") "_3bitrd_3")) ;; Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) (theorem "_3bitrrd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3bitrd_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3bitrd_2" ("wi" "ph" ("wb" "ch" "th"))) ("_3bitrd_3" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" "ta" "ps")) ("bitr3d" "ph" "th" "ta" "ps" "_3bitrd_3" ("bitr2d" "ph" "ps" "ch" "th" "_3bitrd_1" "_3bitrd_2"))) ;; Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) (theorem "_3bitr2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3bitr2d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3bitr2d_2" ("wi" "ph" ("wb" "th" "ch"))) ("_3bitr2d_3" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" "ps" "ta")) ("bitrd" "ph" "ps" "th" "ta" ("bitr4d" "ph" "ps" "ch" "th" "_3bitr2d_1" "_3bitr2d_2") "_3bitr2d_3")) ;; Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) (theorem "_3bitr2rd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3bitr2d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3bitr2d_2" ("wi" "ph" ("wb" "th" "ch"))) ("_3bitr2d_3" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" "ta" "ps")) ("bitr2d" "ph" "ps" "th" "ta" ("bitr4d" "ph" "ps" "ch" "th" "_3bitr2d_1" "_3bitr2d_2") "_3bitr2d_3")) ;; Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.) (theorem "_3bitr3d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3bitr3d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3bitr3d_2" ("wi" "ph" ("wb" "ps" "th"))) ("_3bitr3d_3" ("wi" "ph" ("wb" "ch" "ta")))) (for) ("wi" "ph" ("wb" "th" "ta")) ("bitrd" "ph" "th" "ch" "ta" ("bitr3d" "ph" "ps" "th" "ch" "_3bitr3d_2" "_3bitr3d_1") "_3bitr3d_3")) ;; Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) (theorem "_3bitr3rd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3bitr3d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3bitr3d_2" ("wi" "ph" ("wb" "ps" "th"))) ("_3bitr3d_3" ("wi" "ph" ("wb" "ch" "ta")))) (for) ("wi" "ph" ("wb" "ta" "th")) ("bitr3d" "ph" "ch" "ta" "th" "_3bitr3d_3" ("bitr3d" "ph" "ps" "ch" "th" "_3bitr3d_1" "_3bitr3d_2"))) ;; Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.) (theorem "_3bitr4d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3bitr4d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3bitr4d_2" ("wi" "ph" ("wb" "th" "ps"))) ("_3bitr4d_3" ("wi" "ph" ("wb" "ta" "ch")))) (for) ("wi" "ph" ("wb" "th" "ta")) ("bitrd" "ph" "th" "ps" "ta" "_3bitr4d_2" ("bitr4d" "ph" "ps" "ch" "ta" "_3bitr4d_1" "_3bitr4d_3"))) ;; Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) (theorem "_3bitr4rd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3bitr4d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3bitr4d_2" ("wi" "ph" ("wb" "th" "ps"))) ("_3bitr4d_3" ("wi" "ph" ("wb" "ta" "ch")))) (for) ("wi" "ph" ("wb" "ta" "th")) ("bitr4d" "ph" "ta" "ps" "th" ("bitr4d" "ph" "ta" "ch" "ps" "_3bitr4d_3" "_3bitr4d_1") "_3bitr4d_2")) ;; More general version of ~ 3bitr3i . Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.) (theorem "_3bitr3g" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3bitr3g_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3bitr3g_2" ("wb" "ps" "th")) ("_3bitr3g_3" ("wb" "ch" "ta"))) (for) ("wi" "ph" ("wb" "th" "ta")) ("syl6bb" "ph" "th" "ch" "ta" ("syl5bbr" "th" "ps" "ph" "ch" "_3bitr3g_2" "_3bitr3g_1") "_3bitr3g_3")) ;; More general version of ~ 3bitr4i . Useful for converting definitions in a formula. (Contributed by NM, 11-May-1993.) (theorem "_3bitr4g" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3bitr4g_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3bitr4g_2" ("wb" "th" "ps")) ("_3bitr4g_3" ("wb" "ta" "ch"))) (for) ("wi" "ph" ("wb" "th" "ta")) ("syl6bbr" "ph" "th" "ch" "ta" ("syl5bb" "th" "ps" "ph" "ch" "_3bitr4g_2" "_3bitr4g_1") "_3bitr4g_3")) ;; Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-1993.) (theorem "notnotb" (for ("ph" ( "wff"))) (for) (for) ("wb" "ph" ("wn" ("wn" "ph"))) ("impbii" "ph" ("wn" ("wn" "ph")) ("notnot" "ph") ("notnotr" "ph"))) ;; A biconditional form of contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed by NM, 11-May-1993.) (theorem "con34b" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" "ps") ("wi" ("wn" "ps") ("wn" "ph"))) ("impbii" ("wi" "ph" "ps") ("wi" ("wn" "ps") ("wn" "ph")) ("con3" "ph" "ps") ("con4" "ps" "ph"))) ;; A contraposition deduction. (Contributed by NM, 21-May-1994.) (theorem "con4bid" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("con4bid_1" ("wi" "ph" ("wb" ("wn" "ps") ("wn" "ch"))))) (for) ("wi" "ph" ("wb" "ps" "ch")) ("impcon4bid" "ph" "ps" "ch" ("con4d" "ph" "ch" "ps" ("biimprd" "ph" ("wn" "ps") ("wn" "ch") "con4bid_1")) ("biimpd" "ph" ("wn" "ps") ("wn" "ch") "con4bid_1"))) ;; Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.) (theorem "notbid" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("notbid_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("wn" "ps") ("wn" "ch"))) ("con4bid" "ph" ("wn" "ps") ("wn" "ch") ("_3bitr3g" "ph" "ps" "ch" ("wn" ("wn" "ps")) ("wn" ("wn" "ch")) "notbid_1" ("notnotb" "ps") ("notnotb" "ch")))) ;; Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.) (theorem "notbi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" "ph" "ps") ("wb" ("wn" "ph") ("wn" "ps"))) ("impbii" ("wb" "ph" "ps") ("wb" ("wn" "ph") ("wn" "ps")) ("notbid" ("wb" "ph" "ps") "ph" "ps" ("id" ("wb" "ph" "ps"))) ("con4bid" ("wb" ("wn" "ph") ("wn" "ps")) "ph" "ps" ("id" ("wb" ("wn" "ph") ("wn" "ps")))))) ;; Negate both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) (theorem "notbii" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("notbii_1" ("wb" "ph" "ps"))) (for) ("wb" ("wn" "ph") ("wn" "ps")) ("mpbi" ("wb" "ph" "ps") ("wb" ("wn" "ph") ("wn" "ps")) "notbii_1" ("notbi" "ph" "ps"))) ;; A contraposition inference. (Contributed by NM, 21-May-1994.) (theorem "con4bii" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("con4bii_1" ("wb" ("wn" "ph") ("wn" "ps")))) (for) ("wb" "ph" "ps") ("mpbir" ("wb" "ph" "ps") ("wb" ("wn" "ph") ("wn" "ps")) "con4bii_1" ("notbi" "ph" "ps"))) ;; An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) (theorem "mtbi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("mtbi_1" ("wn" "ph")) ("mtbi_2" ("wb" "ph" "ps"))) (for) ("wn" "ps") ("mto" "ps" "ph" "mtbi_1" ("biimpri" "ph" "ps" "mtbi_2"))) ;; An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.) (theorem "mtbir" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("mtbir_1" ("wn" "ps")) ("mtbir_2" ("wb" "ph" "ps"))) (for) ("wn" "ph") ("mtbi" "ps" "ph" "mtbir_1" ("bicomi" "ph" "ps" "mtbir_2"))) ;; A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.) (theorem "mtbid" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mtbid_min" ("wi" "ph" ("wn" "ps"))) ("mtbid_maj" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wn" "ch")) ("mtod" "ph" "ch" "ps" "mtbid_min" ("biimprd" "ph" "ps" "ch" "mtbid_maj"))) ;; A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.) (theorem "mtbird" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mtbird_min" ("wi" "ph" ("wn" "ch"))) ("mtbird_maj" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wn" "ps")) ("mtod" "ph" "ps" "ch" "mtbird_min" ("biimpd" "ph" "ps" "ch" "mtbird_maj"))) ;; An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.) (theorem "mtbii" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mtbii_min" ("wn" "ps")) ("mtbii_maj" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wn" "ch")) ("mtoi" "ph" "ch" "ps" "mtbii_min" ("biimprd" "ph" "ps" "ch" "mtbii_maj"))) ;; An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.) (theorem "mtbiri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mtbiri_min" ("wn" "ch")) ("mtbiri_maj" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wn" "ps")) ("mtoi" "ph" "ps" "ch" "mtbiri_min" ("biimpd" "ph" "ps" "ch" "mtbiri_maj"))) ;; A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.) (theorem "sylnib" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylnib_1" ("wi" "ph" ("wn" "ps"))) ("sylnib_2" ("wb" "ps" "ch"))) (for) ("wi" "ph" ("wn" "ch")) ("mtbid" "ph" "ps" "ch" "sylnib_1" ("a1i" ("wb" "ps" "ch") "ph" "sylnib_2"))) ;; A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.) (theorem "sylnibr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylnibr_1" ("wi" "ph" ("wn" "ps"))) ("sylnibr_2" ("wb" "ch" "ps"))) (for) ("wi" "ph" ("wn" "ch")) ("sylnib" "ph" "ps" "ch" "sylnibr_1" ("bicomi" "ch" "ps" "sylnibr_2"))) ;; A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.) (theorem "sylnbi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylnbi_1" ("wb" "ph" "ps")) ("sylnbi_2" ("wi" ("wn" "ps") "ch"))) (for) ("wi" ("wn" "ph") "ch") ("sylbi" ("wn" "ph") ("wn" "ps") "ch" ("notbii" "ph" "ps" "sylnbi_1") "sylnbi_2")) ;; A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.) (theorem "sylnbir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("sylnbir_1" ("wb" "ps" "ph")) ("sylnbir_2" ("wi" ("wn" "ps") "ch"))) (for) ("wi" ("wn" "ph") "ch") ("sylnbi" "ph" "ps" "ch" ("bicomi" "ps" "ph" "sylnbir_1") "sylnbir_2")) ;; Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.) (theorem "xchnxbi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("xchnxbi_1" ("wb" ("wn" "ph") "ps")) ("xchnxbi_2" ("wb" "ph" "ch"))) (for) ("wb" ("wn" "ch") "ps") ("bitr3i" ("wn" "ch") ("wn" "ph") "ps" ("notbii" "ph" "ch" "xchnxbi_2") "xchnxbi_1")) ;; Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.) (theorem "xchnxbir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("xchnxbir_1" ("wb" ("wn" "ph") "ps")) ("xchnxbir_2" ("wb" "ch" "ph"))) (for) ("wb" ("wn" "ch") "ps") ("xchnxbi" "ph" "ps" "ch" "xchnxbir_1" ("bicomi" "ch" "ph" "xchnxbir_2"))) ;; Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.) (theorem "xchbinx" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("xchbinx_1" ("wb" "ph" ("wn" "ps"))) ("xchbinx_2" ("wb" "ps" "ch"))) (for) ("wb" "ph" ("wn" "ch")) ("bitri" "ph" ("wn" "ps") ("wn" "ch") "xchbinx_1" ("notbii" "ps" "ch" "xchbinx_2"))) ;; Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.) (theorem "xchbinxr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("xchbinxr_1" ("wb" "ph" ("wn" "ps"))) ("xchbinxr_2" ("wb" "ch" "ps"))) (for) ("wb" "ph" ("wn" "ch")) ("xchbinx" "ph" "ps" "ch" "xchbinxr_1" ("bicomi" "ch" "ps" "xchbinxr_2"))) ;; Introduce an antecedent to both sides of a logical equivalence. This and the next three rules are useful for building up wff's around a definition, in order to make use of the definition. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.) (theorem "imbi2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("imbi2i_1" ("wb" "ph" "ps"))) (for) ("wb" ("wi" "ch" "ph") ("wi" "ch" "ps")) ("pm5_74i" "ch" "ph" "ps" ("a1i" ("wb" "ph" "ps") "ch" "imbi2i_1"))) ;; Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.) (theorem "bibi2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("bibi2i_1" ("wb" "ph" "ps"))) (for) ("wb" ("wb" "ch" "ph") ("wb" "ch" "ps")) ("impbii" ("wb" "ch" "ph") ("wb" "ch" "ps") ("syl6bb" ("wb" "ch" "ph") "ch" "ph" "ps" ("id" ("wb" "ch" "ph")) "bibi2i_1") ("syl6bbr" ("wb" "ch" "ps") "ch" "ps" "ph" ("id" ("wb" "ch" "ps")) "bibi2i_1"))) ;; Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 26-May-1993.) (theorem "bibi1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("bibi2i_1" ("wb" "ph" "ps"))) (for) ("wb" ("wb" "ph" "ch") ("wb" "ps" "ch")) ("_3bitri" ("wb" "ph" "ch") ("wb" "ch" "ph") ("wb" "ch" "ps") ("wb" "ps" "ch") ("bicom" "ph" "ch") ("bibi2i" "ph" "ps" "ch" "bibi2i_1") ("bicom" "ch" "ps"))) ;; The equivalence of two equivalences. (Contributed by NM, 26-May-1993.) (theorem "bibi12i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("bibi2i_1" ("wb" "ph" "ps")) ("bibi12i_2" ("wb" "ch" "th"))) (for) ("wb" ("wb" "ph" "ch") ("wb" "ps" "th")) ("bitri" ("wb" "ph" "ch") ("wb" "ph" "th") ("wb" "ps" "th") ("bibi2i" "ch" "th" "ph" "bibi12i_2") ("bibi1i" "ph" "ps" "th" "bibi2i_1"))) ;; Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) (theorem "imbi2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("imbid_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("wi" "th" "ps") ("wi" "th" "ch"))) ("pm5_74d" "ph" "th" "ps" "ch" ("a1d" "ph" ("wb" "ps" "ch") "th" "imbid_1"))) ;; Deduction adding a consequent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.) (theorem "imbi1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("imbid_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("wi" "ps" "th") ("wi" "ch" "th"))) ("impbid" "ph" ("wi" "ps" "th") ("wi" "ch" "th") ("imim1d" "ph" "ch" "ps" "th" ("biimprd" "ph" "ps" "ch" "imbid_1")) ("imim1d" "ph" "ps" "ch" "th" ("biimpd" "ph" "ps" "ch" "imbid_1")))) ;; Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) (theorem "bibi2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("imbid_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("wb" "th" "ps") ("wb" "th" "ch"))) ("pm5_74ri" "ph" ("wb" "th" "ps") ("wb" "th" "ch") ("_3bitr4i" ("wb" ("wi" "ph" "th") ("wi" "ph" "ps")) ("wb" ("wi" "ph" "th") ("wi" "ph" "ch")) ("wi" "ph" ("wb" "th" "ps")) ("wi" "ph" ("wb" "th" "ch")) ("bibi2i" ("wi" "ph" "ps") ("wi" "ph" "ch") ("wi" "ph" "th") ("pm5_74i" "ph" "ps" "ch" "imbid_1")) ("pm5_74" "ph" "th" "ps") ("pm5_74" "ph" "th" "ch")))) ;; Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 11-May-1993.) (theorem "bibi1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("imbid_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("wb" "ps" "th") ("wb" "ch" "th"))) ("_3bitr4g" "ph" ("wb" "th" "ps") ("wb" "th" "ch") ("wb" "ps" "th") ("wb" "ch" "th") ("bibi2d" "ph" "ps" "ch" "th" "imbid_1") ("bicom" "ps" "th") ("bicom" "ch" "th"))) ;; Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 16-May-1993.) (theorem "imbi12d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imbi12d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("imbi12d_2" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" ("wi" "ps" "th") ("wi" "ch" "ta"))) ("bitrd" "ph" ("wi" "ps" "th") ("wi" "ch" "th") ("wi" "ch" "ta") ("imbi1d" "ph" "ps" "ch" "th" "imbi12d_1") ("imbi2d" "ph" "th" "ta" "ch" "imbi12d_2"))) ;; Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 26-May-1993.) (theorem "bibi12d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imbi12d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("imbi12d_2" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" ("wb" "ps" "th") ("wb" "ch" "ta"))) ("bitrd" "ph" ("wb" "ps" "th") ("wb" "ch" "th") ("wb" "ch" "ta") ("bibi1d" "ph" "ps" "ch" "th" "imbi12d_1") ("bibi2d" "ph" "th" "ta" "ch" "imbi12d_2"))) ;; Closed form of ~ imbi12i . Was automatically derived from its "Virtual Deduction" version and Metamath's "minimize" command. (Contributed by Alan Sare, 18-Mar-2012.) (theorem "imbi12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wi" ("wb" "ch" "th") ("wb" ("wi" "ph" "ch") ("wi" "ps" "th")))) ("expi" ("wb" "ph" "ps") ("wb" "ch" "th") ("wb" ("wi" "ph" "ch") ("wi" "ps" "th")) ("imbi12d" ("wn" ("wi" ("wb" "ph" "ps") ("wn" ("wb" "ch" "th")))) "ph" "ps" "ch" "th" ("simplim" ("wb" "ph" "ps") ("wn" ("wb" "ch" "th"))) ("simprim" ("wb" "ph" "ps") ("wb" "ch" "th"))))) ;; Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (theorem "imbi1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wb" ("wi" "ph" "ch") ("wi" "ps" "ch"))) ("imbi1d" ("wb" "ph" "ps") "ph" "ps" "ch" ("id" ("wb" "ph" "ps")))) ;; Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.) (theorem "imbi2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wb" ("wi" "ch" "ph") ("wi" "ch" "ps"))) ("imbi2d" ("wb" "ph" "ps") "ph" "ps" "ch" ("id" ("wb" "ph" "ps")))) ;; Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.) (theorem "imbi1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("imbi1i_1" ("wb" "ph" "ps"))) (for) ("wb" ("wi" "ph" "ch") ("wi" "ps" "ch")) ("ax_mp" ("wb" "ph" "ps") ("wb" ("wi" "ph" "ch") ("wi" "ps" "ch")) "imbi1i_1" ("imbi1" "ph" "ps" "ch"))) ;; Join two logical equivalences to form equivalence of implications. (Contributed by NM, 1-Aug-1993.) (theorem "imbi12i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("imbi12i_1" ("wb" "ph" "ps")) ("imbi12i_2" ("wb" "ch" "th"))) (for) ("wb" ("wi" "ph" "ch") ("wi" "ps" "th")) ("mp2" ("wb" "ph" "ps") ("wb" "ch" "th") ("wb" ("wi" "ph" "ch") ("wi" "ps" "th")) "imbi12i_1" "imbi12i_2" ("imbi12" "ph" "ps" "ch" "th"))) ;; Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (theorem "bibi1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wb" ("wb" "ph" "ch") ("wb" "ps" "ch"))) ("bibi1d" ("wb" "ph" "ps") "ph" "ps" "ch" ("id" ("wb" "ph" "ps")))) ;; Closed nested implication form of ~ bitr3i . Derived automatically from ~ bitr3VD . (Contributed by Alan Sare, 31-Dec-2011.) (theorem "bitr3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wi" ("wb" "ph" "ch") ("wb" "ps" "ch"))) ("biimpd" ("wb" "ph" "ps") ("wb" "ph" "ch") ("wb" "ps" "ch") ("bibi1" "ph" "ps" "ch"))) ;; Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117. (Contributed by NM, 15-Apr-1995.) (Proof shortened by Wolf Lammen, 3-Jan-2013.) (theorem "con2bi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" "ph" ("wn" "ps")) ("wb" "ps" ("wn" "ph"))) ("_3bitr2i" ("wb" "ph" ("wn" "ps")) ("wb" ("wn" "ph") ("wn" ("wn" "ps"))) ("wb" ("wn" "ph") "ps") ("wb" "ps" ("wn" "ph")) ("notbi" "ph" ("wn" "ps")) ("bibi2i" "ps" ("wn" ("wn" "ps")) ("wn" "ph") ("notnotb" "ps")) ("bicom" ("wn" "ph") "ps"))) ;; A contraposition deduction. (Contributed by NM, 15-Apr-1995.) (theorem "con2bid" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("con2bid_1" ("wi" "ph" ("wb" "ps" ("wn" "ch"))))) (for) ("wi" "ph" ("wb" "ch" ("wn" "ps"))) ("sylibr" "ph" ("wb" "ps" ("wn" "ch")) ("wb" "ch" ("wn" "ps")) "con2bid_1" ("con2bi" "ch" "ps"))) ;; A contraposition deduction. (Contributed by NM, 9-Oct-1999.) (theorem "con1bid" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("con1bid_1" ("wi" "ph" ("wb" ("wn" "ps") "ch")))) (for) ("wi" "ph" ("wb" ("wn" "ch") "ps")) ("bicomd" "ph" "ps" ("wn" "ch") ("con2bid" "ph" "ch" "ps" ("bicomd" "ph" ("wn" "ps") "ch" "con1bid_1")))) ;; A contraposition inference. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) (theorem "con1bii" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("con1bii_1" ("wb" ("wn" "ph") "ps"))) (for) ("wb" ("wn" "ps") "ph") ("bicomi" "ph" ("wn" "ps") ("xchbinx" "ph" ("wn" "ph") "ps" ("notnotb" "ph") "con1bii_1"))) ;; A contraposition inference. (Contributed by NM, 12-Mar-1993.) (theorem "con2bii" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("con2bii_1" ("wb" "ph" ("wn" "ps")))) (for) ("wb" "ps" ("wn" "ph")) ("bicomi" ("wn" "ph") "ps" ("con1bii" "ps" "ph" ("bicomi" "ph" ("wn" "ps") "con2bii_1")))) ;; Contraposition. Bidirectional version of ~ con1 . (Contributed by NM, 3-Jan-1993.) (theorem "con1b" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" ("wn" "ph") "ps") ("wi" ("wn" "ps") "ph")) ("impbii" ("wi" ("wn" "ph") "ps") ("wi" ("wn" "ps") "ph") ("con1" "ph" "ps") ("con1" "ps" "ph"))) ;; Contraposition. Bidirectional version of ~ con2 . (Contributed by NM, 12-Mar-1993.) (theorem "con2b" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wn" "ps")) ("wi" "ps" ("wn" "ph"))) ("impbii" ("wi" "ph" ("wn" "ps")) ("wi" "ps" ("wn" "ph")) ("con2" "ph" "ps") ("con2" "ps" "ph"))) ;; A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.) (theorem "biimt" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wb" "ps" ("wi" "ph" "ps"))) ("impbid2" "ph" "ps" ("wi" "ph" "ps") ("ax_1" "ps" "ph") ("pm2_27" "ph" "ps"))) ;; Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_5" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wb" ("wi" "ph" "ps") "ps")) ("bicomd" "ph" "ps" ("wi" "ph" "ps") ("biimt" "ph" "ps"))) ;; Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) (theorem "a1bi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("a1bi_1" "ph")) (for) ("wb" "ps" ("wi" "ph" "ps")) ("ax_mp" "ph" ("wb" "ps" ("wi" "ph" "ps")) "a1bi_1" ("biimt" "ph" "ps"))) ;; A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) (theorem "mt2bi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("mt2bi_1" "ph")) (for) ("wb" ("wn" "ps") ("wi" "ps" ("wn" "ph"))) ("bitri" ("wn" "ps") ("wi" "ph" ("wn" "ps")) ("wi" "ps" ("wn" "ph")) ("a1bi" "ph" ("wn" "ps") "mt2bi_1") ("con2b" "ph" "ps"))) ;; Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) (theorem "mtt" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wb" ("wn" "ps") ("wi" "ps" "ph"))) ("syl6bbr" ("wn" "ph") ("wn" "ps") ("wi" ("wn" "ph") ("wn" "ps")) ("wi" "ps" "ph") ("biimt" ("wn" "ph") ("wn" "ps")) ("con34b" "ps" "ph"))) ;; If a proposition is false, then implying it is equivalent to being false. One of four theorems that can be used to simplify an implication ` ( ph -> ps ) ` , the other ones being ~ ax-1 (true consequent), ~ pm2.21 (false antecedent), ~ pm5.5 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019.) (Proof shortened by Wolf Lammen, 26-May-2019.) (theorem "imnot" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" "ps") ("wb" ("wi" "ph" "ps") ("wn" "ph"))) ("bicomd" ("wn" "ps") ("wn" "ph") ("wi" "ph" "ps") ("mtt" "ps" "ph"))) ;; Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_501" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wb" "ps" ("wb" "ph" "ps"))) ("impbid" "ph" "ps" ("wb" "ph" "ps") ("pm5_1im" "ph" "ps") ("com12" ("wb" "ph" "ps") "ph" "ps" ("biimp" "ph" "ps")))) ;; Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.) (theorem "ibib" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" "ps") ("wi" "ph" ("wb" "ph" "ps"))) ("pm5_74i" "ph" "ps" ("wb" "ph" "ps") ("pm5_501" "ph" "ps"))) ;; Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.) (theorem "ibibr" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" "ps") ("wi" "ph" ("wb" "ps" "ph"))) ("pm5_74i" "ph" "ps" ("wb" "ps" "ph") ("syl6bb" "ph" "ps" ("wb" "ph" "ps") ("wb" "ps" "ph") ("pm5_501" "ph" "ps") ("bicom" "ph" "ps")))) ;; A wff is equivalent to its equivalence with a truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "tbt" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("tbt_1" "ph")) (for) ("wb" "ps" ("wb" "ps" "ph")) ("ax_mp" "ph" ("wb" "ps" ("wb" "ps" "ph")) "tbt_1" ("pm5_74ri" "ph" "ps" ("wb" "ps" "ph") ("ibibr" "ph" "ps")))) ;; The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Proof shortened by Wolf Lammen, 28-Jan-2013.) (theorem "nbn2" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wb" ("wn" "ps") ("wb" "ph" "ps"))) ("syl6bbr" ("wn" "ph") ("wn" "ps") ("wb" ("wn" "ph") ("wn" "ps")) ("wb" "ph" "ps") ("pm5_501" ("wn" "ph") ("wn" "ps")) ("notbi" "ph" "ps"))) ;; Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.) (theorem "bibif" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" "ps") ("wb" ("wb" "ph" "ps") ("wn" "ph"))) ("syl6rbb" ("wn" "ps") ("wn" "ph") ("wb" "ps" "ph") ("wb" "ph" "ps") ("nbn2" "ps" "ph") ("bicom" "ps" "ph"))) ;; The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) (theorem "nbn" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("nbn_1" ("wn" "ph"))) (for) ("wb" ("wn" "ps") ("wb" "ps" "ph")) ("bicomi" ("wb" "ps" "ph") ("wn" "ps") ("ax_mp" ("wn" "ph") ("wb" ("wb" "ps" "ph") ("wn" "ps")) "nbn_1" ("bibif" "ps" "ph")))) ;; Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.) (theorem "nbn3" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("nbn3_1" "ph")) (for) ("wb" ("wn" "ps") ("wb" "ps" ("wn" "ph"))) ("nbn" ("wn" "ph") "ps" ("notnoti" "ph" "nbn3_1"))) ;; Two propositions are equivalent if they are both false. Closed form of ~ 2false . Equivalent to a ~ biimpr -like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) (theorem "pm5_21im" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wi" ("wn" "ps") ("wb" "ph" "ps"))) ("biimpd" ("wn" "ph") ("wn" "ps") ("wb" "ph" "ps") ("nbn2" "ph" "ps"))) ;; Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.) (theorem "_2false" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("_2false_1" ("wn" "ph")) ("_2false_2" ("wn" "ps"))) (for) ("wb" "ph" "ps") ("con4bii" "ph" "ps" ("_2th" ("wn" "ph") ("wn" "ps") "_2false_1" "_2false_2"))) ;; Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.) (theorem "_2falsed" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_2falsed_1" ("wi" "ph" ("wn" "ps"))) ("_2falsed_2" ("wi" "ph" ("wn" "ch")))) (for) ("wi" "ph" ("wb" "ps" "ch")) ("impbid" "ph" "ps" "ch" ("pm2_21d" "ph" "ps" "ch" "_2falsed_1") ("pm2_21d" "ph" "ch" "ps" "_2falsed_2"))) ;; Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.) (theorem "pm5_21ni" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm5_21ni_1" ("wi" "ph" "ps")) ("pm5_21ni_2" ("wi" "ch" "ps"))) (for) ("wi" ("wn" "ps") ("wb" "ph" "ch")) ("_2falsed" ("wn" "ps") "ph" "ch" ("con3i" "ph" "ps" "pm5_21ni_1") ("con3i" "ch" "ps" "pm5_21ni_2"))) ;; Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.) (theorem "pm5_21nii" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm5_21ni_1" ("wi" "ph" "ps")) ("pm5_21ni_2" ("wi" "ch" "ps")) ("pm5_21nii_3" ("wi" "ps" ("wb" "ph" "ch")))) (for) ("wb" "ph" "ch") ("pm2_61i" "ps" ("wb" "ph" "ch") "pm5_21nii_3" ("pm5_21ni" "ph" "ps" "ch" "pm5_21ni_1" "pm5_21ni_2"))) ;; Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Proof shortened by Wolf Lammen, 6-Oct-2013.) (theorem "pm5_21ndd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm5_21ndd_1" ("wi" "ph" ("wi" "ch" "ps"))) ("pm5_21ndd_2" ("wi" "ph" ("wi" "th" "ps"))) ("pm5_21ndd_3" ("wi" "ph" ("wi" "ps" ("wb" "ch" "th"))))) (for) ("wi" "ph" ("wb" "ch" "th")) ("pm2_61d" "ph" "ps" ("wb" "ch" "th") "pm5_21ndd_3" ("syl6c" "ph" ("wn" "ps") ("wn" "ch") ("wn" "th") ("wb" "ch" "th") ("con3d" "ph" "ch" "ps" "pm5_21ndd_1") ("con3d" "ph" "th" "ps" "pm5_21ndd_2") ("pm5_21im" "ch" "th")))) ;; Combine antecedents into a single biconditional. This inference, reminiscent of ~ ja , is reversible: The hypotheses can be deduced from the conclusion alone (see ~ pm5.1im and ~ pm5.21im ). (Contributed by Wolf Lammen, 13-May-2013.) (theorem "bija" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("bija_1" ("wi" "ph" ("wi" "ps" "ch"))) ("bija_2" ("wi" ("wn" "ph") ("wi" ("wn" "ps") "ch")))) (for) ("wi" ("wb" "ph" "ps") "ch") ("pm2_61d" ("wb" "ph" "ps") "ps" "ch" ("syli" "ps" ("wb" "ph" "ps") "ph" "ch" ("biimpr" "ph" "ps") "bija_1") ("syli" ("wn" "ps") ("wb" "ph" "ps") ("wn" "ph") "ch" ("con3d" ("wb" "ph" "ps") "ph" "ps" ("biimp" "ph" "ps")) "bija_2"))) ;; Theorem *5.18 of [WhiteheadRussell] p. 124. This theorem says that logical equivalence is the same as negated "exclusive-or." (Contributed by NM, 28-Jun-2002.) (Proof shortened by Andrew Salmon, 20-Jun-2011.) (Proof shortened by Wolf Lammen, 15-Oct-2013.) (theorem "pm5_18" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" "ph" "ps") ("wn" ("wb" "ph" ("wn" "ps")))) ("pm2_61i" "ph" ("wb" ("wb" "ph" "ps") ("wn" ("wb" "ph" ("wn" "ps")))) ("bitr2d" "ph" ("wn" ("wb" "ph" ("wn" "ps"))) "ps" ("wb" "ph" "ps") ("con1bid" "ph" "ps" ("wb" "ph" ("wn" "ps")) ("pm5_501" "ph" ("wn" "ps"))) ("pm5_501" "ph" "ps")) ("bitr2d" ("wn" "ph") ("wn" ("wb" "ph" ("wn" "ps"))) ("wn" "ps") ("wb" "ph" "ps") ("con1bid" ("wn" "ph") ("wn" "ps") ("wb" "ph" ("wn" "ps")) ("nbn2" "ph" ("wn" "ps"))) ("nbn2" "ph" "ps")))) ;; Two ways to express "exclusive or." (Contributed by NM, 1-Jan-2006.) (theorem "xor3" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wb" "ph" "ps")) ("wb" "ph" ("wn" "ps"))) ("bicomi" ("wb" "ph" ("wn" "ps")) ("wn" ("wb" "ph" "ps")) ("con2bii" ("wb" "ph" "ps") ("wb" "ph" ("wn" "ps")) ("pm5_18" "ph" "ps")))) ;; Move negation outside of biconditional. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 20-Sep-2013.) (theorem "nbbn" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" ("wn" "ph") "ps") ("wn" ("wb" "ph" "ps"))) ("_3bitrri" ("wn" ("wb" "ph" "ps")) ("wb" "ph" ("wn" "ps")) ("wb" "ps" ("wn" "ph")) ("wb" ("wn" "ph") "ps") ("xor3" "ph" "ps") ("con2bi" "ph" "ps") ("bicom" "ps" ("wn" "ph")))) ;; Associative law for the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure and Applied Logic, 113:207-224, 2002, ~ http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805 . Interestingly, this law was not included in _Principia Mathematica_ but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by NM, 8-Jan-2005.) (Proof shortened by Juha Arpiainen, 19-Jan-2006.) (Proof shortened by Wolf Lammen, 21-Sep-2013.) (theorem "biass" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wb" ("wb" "ph" "ps") "ch") ("wb" "ph" ("wb" "ps" "ch"))) ("pm2_61i" "ph" ("wb" ("wb" ("wb" "ph" "ps") "ch") ("wb" "ph" ("wb" "ps" "ch"))) ("bitr3d" "ph" ("wb" "ps" "ch") ("wb" ("wb" "ph" "ps") "ch") ("wb" "ph" ("wb" "ps" "ch")) ("bibi1d" "ph" "ps" ("wb" "ph" "ps") "ch" ("pm5_501" "ph" "ps")) ("pm5_501" "ph" ("wb" "ps" "ch"))) ("bitr3d" ("wn" "ph") ("wn" ("wb" "ps" "ch")) ("wb" ("wb" "ph" "ps") "ch") ("wb" "ph" ("wb" "ps" "ch")) ("syl5bbr" ("wn" ("wb" "ps" "ch")) ("wb" ("wn" "ps") "ch") ("wn" "ph") ("wb" ("wb" "ph" "ps") "ch") ("nbbn" "ps" "ch") ("bibi1d" ("wn" "ph") ("wn" "ps") ("wb" "ph" "ps") "ch" ("nbn2" "ph" "ps"))) ("nbn2" "ph" ("wb" "ps" "ch"))))) ;; Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_19" (for ("ph" ( "wff"))) (for) (for) ("wn" ("wb" "ph" ("wn" "ph"))) ("mpbi" ("wb" "ph" "ph") ("wn" ("wb" "ph" ("wn" "ph"))) ("biid" "ph") ("pm5_18" "ph" "ph"))) ;; Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 11-May-1993.) (theorem "bi2_04" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ps" ("wi" "ph" "ch"))) ("impbii" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ps" ("wi" "ph" "ch")) ("pm2_04" "ph" "ps" "ch") ("pm2_04" "ps" "ph" "ch"))) ;; Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.) (theorem "pm5_4" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wi" "ph" "ps")) ("wi" "ph" "ps")) ("impbii" ("wi" "ph" ("wi" "ph" "ps")) ("wi" "ph" "ps") ("pm2_43" "ph" "ps") ("ax_1" ("wi" "ph" "ps") "ph"))) ;; Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.) (theorem "imdi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wi" "ps" "ch")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch"))) ("impbii" ("wi" "ph" ("wi" "ps" "ch")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("ax_2" "ph" "ps" "ch") ("pm2_86" "ph" "ps" "ch"))) ;; Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.) (theorem "pm5_41" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("wi" "ph" ("wi" "ps" "ch"))) ("bicomi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("imdi" "ph" "ps" "ch"))) ;; Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_8" (for ("ph" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wn" "ph")) ("wn" "ph")) ("impbii" ("wi" "ph" ("wn" "ph")) ("wn" "ph") ("pm2_01" "ph") ("ax_1" ("wn" "ph") "ph"))) ;; Theorem *4.81 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_81" (for ("ph" ( "wff"))) (for) (for) ("wb" ("wi" ("wn" "ph") "ph") "ph") ("impbii" ("wi" ("wn" "ph") "ph") "ph" ("pm2_18" "ph") ("pm2_24" "ph" "ph"))) ;; Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Wolf Lammen, 14-Sep-2013.) (theorem "imim21b" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wi" "ps" "ph") ("wb" ("wi" ("wi" "ph" "ch") ("wi" "ps" "th")) ("wi" "ps" ("wi" "ch" "th")))) ("syl5bb" ("wi" ("wi" "ph" "ch") ("wi" "ps" "th")) ("wi" "ps" ("wi" ("wi" "ph" "ch") "th")) ("wi" "ps" "ph") ("wi" "ps" ("wi" "ch" "th")) ("bi2_04" ("wi" "ph" "ch") "ps" "th") ("pm5_74d" ("wi" "ps" "ph") "ps" ("wi" ("wi" "ph" "ch") "th") ("wi" "ch" "th") ("imim2i" "ph" ("wb" ("wi" ("wi" "ph" "ch") "th") ("wi" "ch" "th")) "ps" ("imbi1d" "ph" ("wi" "ph" "ch") "ch" "th" ("pm5_5" "ph" "ch")))))) ;; Extend wff definition to include disjunction ('or'). (term "wo" ( ( "wff") ( "wff") ( "wff"))) ;; Extend wff definition to include conjunction ('and'). (term "wa" ( ( "wff") ( "wff") ( "wff"))) ;; Define disjunction (logical 'or'). Definition of [Margaris] p. 49. When the left operand, right operand, or both are true, the result is true; when both sides are false, the result is false. For example, it is true that ` ( 2 = 3 \/ 4 = 4 ) ` ( ~ ex-or ). After we define the constant true ` T. ` ( ~ df-tru ) and the constant false ` F. ` ( ~ df-fal ), we will be able to prove these truth table values: ` ( ( T. \/ T. ) <-> T. ) ` ( ~ truortru ), ` ( ( T. \/ F. ) <-> T. ) ` ( ~ truorfal ), ` ( ( F. \/ T. ) <-> T. ) ` ( ~ falortru ), and ` ( ( F. \/ F. ) <-> F. ) ` ( ~ falorfal ). This is our first use of the biconditional connective in a definition; we use the biconditional connective in place of the traditional "<=def=>", which means the same thing, except that we can manipulate the biconditional connective directly in proofs rather than having to rely on an informal definition substitution rule. Note that if we mechanically substitute ` ( -. ph -> ps ) ` for ` ( ph \/ ps ) ` , we end up with an instance of previously proved theorem ~ biid . This is the justification for the definition, along with the fact that it introduces a new symbol ` \/ ` . Contrast with ` /\ ` ( ~ df-an ), ` -> ` ( ~ wi ), ` -/\ ` ( ~ df-nan ), and ` \/_ ` ( ~ df-xor ) . (Contributed by NM, 27-Dec-1992.) (axiom "df_or" (!! ( ("ph" ( "wff")) ("ps" ( "wff"))) ("wb" ("wo" "ph" "ps") ("wi" ("wn" "ph") "ps")))) ;; Define conjunction (logical 'and'). Definition of [Margaris] p. 49. When both the left and right operand are true, the result is true; when either is false, the result is false. For example, it is true that ` ( 2 = 2 /\ 3 = 3 ) ` . After we define the constant true ` T. ` ( ~ df-tru ) and the constant false ` F. ` ( ~ df-fal ), we will be able to prove these truth table values: ` ( ( T. /\ T. ) <-> T. ) ` ( ~ truantru ), ` ( ( T. /\ F. ) <-> F. ) ` ( ~ truanfal ), ` ( ( F. /\ T. ) <-> F. ) ` ( ~ falantru ), and ` ( ( F. /\ F. ) <-> F. ) ` ( ~ falanfal ). Contrast with ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), ` -/\ ` ( ~ df-nan ), and ` \/_ ` ( ~ df-xor ) . (Contributed by NM, 5-Jan-1993.) (axiom "df_an" (!! ( ("ph" ( "wff")) ("ps" ( "wff"))) ("wb" ("wa" "ph" "ps") ("wn" ("wi" "ph" ("wn" "ps")))))) ;; Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_64" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" ("wn" "ph") "ps") ("wo" "ph" "ps")) ("bicomi" ("wo" "ph" "ps") ("wi" ("wn" "ph") "ps") ("df_or" "ph" "ps"))) ;; Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_53" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wo" "ph" "ps") ("wi" ("wn" "ph") "ps")) ("biimpi" ("wo" "ph" "ps") ("wi" ("wn" "ph") "ps") ("df_or" "ph" "ps"))) ;; Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_54" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" ("wn" "ph") "ps") ("wo" "ph" "ps")) ("biimpri" ("wo" "ph" "ps") ("wi" ("wn" "ph") "ps") ("df_or" "ph" "ps"))) ;; Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (theorem "ori" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("ori_1" ("wo" "ph" "ps"))) (for) ("wi" ("wn" "ph") "ps") ("mpbi" ("wo" "ph" "ps") ("wi" ("wn" "ph") "ps") "ori_1" ("df_or" "ph" "ps"))) ;; Infer disjunction from implication. (Contributed by NM, 11-Jun-1994.) (theorem "orri" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("orri_1" ("wi" ("wn" "ph") "ps"))) (for) ("wo" "ph" "ps") ("mpbir" ("wo" "ph" "ps") ("wi" ("wn" "ph") "ps") "orri_1" ("df_or" "ph" "ps"))) ;; Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (theorem "ord" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ord_1" ("wi" "ph" ("wo" "ps" "ch")))) (for) ("wi" "ph" ("wi" ("wn" "ps") "ch")) ("sylib" "ph" ("wo" "ps" "ch") ("wi" ("wn" "ps") "ch") "ord_1" ("df_or" "ps" "ch"))) ;; Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.) (theorem "orrd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("orrd_1" ("wi" "ph" ("wi" ("wn" "ps") "ch")))) (for) ("wi" "ph" ("wo" "ps" "ch")) ("syl" "ph" ("wi" ("wn" "ps") "ch") ("wo" "ps" "ch") "orrd_1" ("pm2_54" "ps" "ch"))) ;; Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (theorem "jaoi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("jaoi_1" ("wi" "ph" "ps")) ("jaoi_2" ("wi" "ch" "ps"))) (for) ("wi" ("wo" "ph" "ch") "ps") ("pm2_61d2" ("wo" "ph" "ch") "ph" "ps" ("syl6" ("wo" "ph" "ch") ("wn" "ph") "ch" "ps" ("pm2_53" "ph" "ch") "jaoi_2") "jaoi_1")) ;; Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) (theorem "jaod" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jaod_1" ("wi" "ph" ("wi" "ps" "ch"))) ("jaod_2" ("wi" "ph" ("wi" "th" "ch")))) (for) ("wi" "ph" ("wi" ("wo" "ps" "th") "ch")) ("com12" ("wo" "ps" "th") "ph" "ch" ("jaoi" "ps" ("wi" "ph" "ch") "th" ("com12" "ph" "ps" "ch" "jaod_1") ("com12" "ph" "th" "ch" "jaod_2")))) ;; Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.) (theorem "mpjaod" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jaod_1" ("wi" "ph" ("wi" "ps" "ch"))) ("jaod_2" ("wi" "ph" ("wi" "th" "ch"))) ("jaod_3" ("wi" "ph" ("wo" "ps" "th")))) (for) ("wi" "ph" "ch") ("mpd" "ph" ("wo" "ps" "th") "ch" "jaod_3" ("jaod" "ph" "ps" "ch" "th" "jaod_1" "jaod_2"))) ;; Elimination of disjunction by denial of a disjunct. Theorem *2.55 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 21-Jul-2012.) (theorem "orel1" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wi" ("wo" "ph" "ps") "ps")) ("com12" ("wo" "ph" "ps") ("wn" "ph") "ps" ("pm2_53" "ph" "ps"))) ;; Elimination of disjunction by denial of a disjunct. Theorem *2.56 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Apr-2013.) (theorem "orel2" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wi" ("wo" "ps" "ph") "ps")) ("jaod" ("wn" "ph") "ps" "ps" "ph" ("idd" ("wn" "ph") "ps") ("pm2_21" "ph" "ps"))) ;; Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) (theorem "olc" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wo" "ps" "ph")) ("orrd" "ph" "ps" "ph" ("ax_1" "ph" ("wn" "ps")))) ;; Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) (theorem "orc" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wo" "ph" "ps")) ("orrd" "ph" "ph" "ps" ("pm2_24" "ph" "ps"))) ;; Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (theorem "pm1_4" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wo" "ph" "ps") ("wo" "ps" "ph")) ("jaoi" "ph" ("wo" "ps" "ph") "ps" ("olc" "ph" "ps") ("orc" "ps" "ph"))) ;; Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.) (theorem "orcom" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wo" "ph" "ps") ("wo" "ps" "ph")) ("impbii" ("wo" "ph" "ps") ("wo" "ps" "ph") ("pm1_4" "ph" "ps") ("pm1_4" "ps" "ph"))) ;; Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.) (theorem "orcomd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("orcomd_1" ("wi" "ph" ("wo" "ps" "ch")))) (for) ("wi" "ph" ("wo" "ch" "ps")) ("sylib" "ph" ("wo" "ps" "ch") ("wo" "ch" "ps") "orcomd_1" ("orcom" "ps" "ch"))) ;; Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.) (theorem "orcoms" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("orcoms_1" ("wi" ("wo" "ph" "ps") "ch"))) (for) ("wi" ("wo" "ps" "ph") "ch") ("syl" ("wo" "ps" "ph") ("wo" "ph" "ps") "ch" ("pm1_4" "ps" "ph") "orcoms_1")) ;; Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) (theorem "orci" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("orci_1" "ph")) (for) ("wo" "ph" "ps") ("orri" "ph" "ps" ("pm2_24i" "ph" "ps" "orci_1"))) ;; Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) (theorem "olci" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("orci_1" "ph")) (for) ("wo" "ps" "ph") ("orri" "ps" "ph" ("a1i" "ph" ("wn" "ps") "orci_1"))) ;; Deduction introducing a disjunct. A translation of natural deduction rule ` \/ ` IR ( ` \/ ` insertion right), see ~ natded . (Contributed by NM, 20-Sep-2007.) (theorem "orcd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("orcd_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("wo" "ps" "ch")) ("syl" "ph" "ps" ("wo" "ps" "ch") "orcd_1" ("orc" "ps" "ch"))) ;; Deduction introducing a disjunct. A translation of natural deduction rule ` \/ ` IL ( ` \/ ` insertion left), see ~ natded . (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) (theorem "olcd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("orcd_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("wo" "ch" "ps")) ("orcomd" "ph" "ps" "ch" ("orcd" "ph" "ps" "ch" "orcd_1"))) ;; Deduction eliminating disjunct. _Notational convention_: We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism ( ~ syl ) -type inference in a proof. (Contributed by NM, 21-Jun-1994.) (theorem "orcs" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("orcs_1" ("wi" ("wo" "ph" "ps") "ch"))) (for) ("wi" "ph" "ch") ("syl" "ph" ("wo" "ph" "ps") "ch" ("orc" "ph" "ps") "orcs_1")) ;; Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) (theorem "olcs" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("olcs_1" ("wi" ("wo" "ph" "ps") "ch"))) (for) ("wi" "ps" "ch") ("orcs" "ps" "ph" "ch" ("orcoms" "ph" "ps" "ch" "olcs_1"))) ;; Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_07" (for ("ph" ( "wff"))) (for) (for) ("wi" "ph" ("wo" "ph" "ph")) ("olc" "ph" "ph")) ;; Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_45" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wo" "ph" "ps")) ("wn" "ph")) ("con3i" "ph" ("wo" "ph" "ps") ("orc" "ph" "ps"))) ;; Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_46" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wo" "ph" "ps")) ("wn" "ps")) ("con3i" "ps" ("wo" "ph" "ps") ("olc" "ps" "ph"))) ;; Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_47" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wo" "ph" "ps")) ("wo" ("wn" "ph") "ps")) ("orcd" ("wn" ("wo" "ph" "ps")) ("wn" "ph") "ps" ("pm2_45" "ph" "ps"))) ;; Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_48" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wo" "ph" "ps")) ("wo" "ph" ("wn" "ps"))) ("olcd" ("wn" ("wo" "ph" "ps")) ("wn" "ps") "ph" ("pm2_46" "ph" "ps"))) ;; Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_49" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wo" "ph" "ps")) ("wo" ("wn" "ph") ("wn" "ps"))) ("olcd" ("wn" ("wo" "ph" "ps")) ("wn" "ps") ("wn" "ph") ("pm2_46" "ph" "ps"))) ;; Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_67_2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" ("wo" "ph" "ch") "ps") ("wi" "ph" "ps")) ("imim1i" "ph" ("wo" "ph" "ch") "ps" ("orc" "ph" "ch"))) ;; Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_67" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" ("wo" "ph" "ps") "ps") ("wi" "ph" "ps")) ("pm2_67_2" "ph" "ps" "ps")) ;; Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_25" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wo" "ph" ("wi" ("wo" "ph" "ps") "ps")) ("orri" "ph" ("wi" ("wo" "ph" "ps") "ps") ("orel1" "ph" "ps"))) ;; A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) (theorem "biorf" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wb" "ps" ("wo" "ph" "ps"))) ("impbid2" ("wn" "ph") "ps" ("wo" "ph" "ps") ("olc" "ps" "ph") ("orel1" "ph" "ps"))) ;; A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.) (theorem "biortn" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wb" "ps" ("wo" ("wn" "ph") "ps"))) ("syl" "ph" ("wn" ("wn" "ph")) ("wb" "ps" ("wo" ("wn" "ph") "ps")) ("notnot" "ph") ("biorf" ("wn" "ph") "ps"))) ;; A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 16-Jul-2021.) (theorem "biorfi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("biorfi_1" ("wn" "ph"))) (for) ("wb" "ps" ("wo" "ps" "ph")) ("impbii" "ps" ("wo" "ps" "ph") ("orc" "ps" "ph") ("ax_mp" ("wn" "ph") ("wi" ("wo" "ps" "ph") "ps") "biorfi_1" ("orel2" "ph" "ps")))) ;; Obsolete proof of ~ biorfi as of 16-Jul-2021. (Contributed by NM, 23-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "biorfiOLD" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("biorfi_1" ("wn" "ph"))) (for) ("wb" "ps" ("wo" "ps" "ph")) ("ax_mp" ("wn" "ph") ("wb" "ps" ("wo" "ps" "ph")) "biorfi_1" ("impbid2" ("wn" "ph") "ps" ("wo" "ps" "ph") ("orc" "ps" "ph") ("orel2" "ph" "ps")))) ;; Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_621" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wo" "ph" "ps") "ps")) ("jaod" ("wi" "ph" "ps") "ph" "ps" "ps" ("id" ("wi" "ph" "ps")) ("idd" ("wi" "ph" "ps") "ps"))) ;; Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.) (theorem "pm2_62" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wo" "ph" "ps") ("wi" ("wi" "ph" "ps") "ps")) ("com12" ("wi" "ph" "ps") ("wo" "ph" "ps") "ps" ("pm2_621" "ph" "ps"))) ;; Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_68" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" ("wi" "ph" "ps") "ps") ("wo" "ph" "ps")) ("orrd" ("wi" ("wi" "ph" "ps") "ps") "ph" "ps" ("jarl" "ph" "ps" "ps"))) ;; Logical 'or' expressed in terms of implication only. Theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Wolf Lammen, 20-Oct-2012.) (theorem "dfor2" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wo" "ph" "ps") ("wi" ("wi" "ph" "ps") "ps")) ("impbii" ("wo" "ph" "ps") ("wi" ("wi" "ph" "ps") "ps") ("pm2_62" "ph" "ps") ("pm2_68" "ph" "ps"))) ;; Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (theorem "imor" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" "ps") ("wo" ("wn" "ph") "ps")) ("bitr4i" ("wi" "ph" "ps") ("wi" ("wn" ("wn" "ph")) "ps") ("wo" ("wn" "ph") "ps") ("imbi1i" "ph" ("wn" ("wn" "ph")) "ps" ("notnotb" "ph")) ("df_or" ("wn" "ph") "ps"))) ;; Infer disjunction from implication. (Contributed by NM, 12-Mar-2012.) (theorem "imori" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("imori_1" ("wi" "ph" "ps"))) (for) ("wo" ("wn" "ph") "ps") ("mpbi" ("wi" "ph" "ps") ("wo" ("wn" "ph") "ps") "imori_1" ("imor" "ph" "ps"))) ;; Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (theorem "imorri" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("imorri_1" ("wo" ("wn" "ph") "ps"))) (for) ("wi" "ph" "ps") ("mpbir" ("wi" "ph" "ps") ("wo" ("wn" "ph") "ps") "imorri_1" ("imor" "ph" "ps"))) ;; Law of excluded middle, also called the principle of _tertium non datur_. Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is either true or not true; there are no in-between values of truth. This is an essential distinction of our classical logic and is not a theorem of intuitionistic logic. In intuitionistic logic, if this statement is true for some ` ph ` , then ` ph ` is decideable. (Contributed by NM, 29-Dec-1992.) (theorem "exmid" (for ("ph" ( "wff"))) (for) (for) ("wo" "ph" ("wn" "ph")) ("orri" "ph" ("wn" "ph") ("id" ("wn" "ph")))) ;; Law of excluded middle in a context. (Contributed by Mario Carneiro, 9-Feb-2017.) (theorem "exmidd" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wo" "ps" ("wn" "ps"))) ("a1i" ("wo" "ps" ("wn" "ps")) "ph" ("exmid" "ps"))) ;; Theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) (theorem "pm2_1" (for ("ph" ( "wff"))) (for) (for) ("wo" ("wn" "ph") "ph") ("imori" "ph" "ph" ("id" "ph"))) ;; Theorem *2.13 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_13" (for ("ph" ( "wff"))) (for) (for) ("wo" "ph" ("wn" ("wn" ("wn" "ph")))) ("orri" "ph" ("wn" ("wn" ("wn" "ph"))) ("notnot" ("wn" "ph")))) ;; Theorem *4.62 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_62" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wn" "ps")) ("wo" ("wn" "ph") ("wn" "ps"))) ("imor" "ph" ("wn" "ps"))) ;; Theorem *4.66 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_66" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" ("wn" "ph") ("wn" "ps")) ("wo" "ph" ("wn" "ps"))) ("pm4_64" "ph" ("wn" "ps"))) ;; Theorem *4.63 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_63" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wi" "ph" ("wn" "ps"))) ("wa" "ph" "ps")) ("bicomi" ("wa" "ph" "ps") ("wn" ("wi" "ph" ("wn" "ps"))) ("df_an" "ph" "ps"))) ;; Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (theorem "imnan" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wn" "ps")) ("wn" ("wa" "ph" "ps"))) ("con2bii" ("wa" "ph" "ps") ("wi" "ph" ("wn" "ps")) ("df_an" "ph" "ps"))) ;; Infer implication from negated conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.) (theorem "imnani" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("imnani_1" ("wn" ("wa" "ph" "ps")))) (for) ("wi" "ph" ("wn" "ps")) ("mpbir" ("wi" "ph" ("wn" "ps")) ("wn" ("wa" "ph" "ps")) "imnani_1" ("imnan" "ph" "ps"))) ;; Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2012.) (theorem "iman" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" "ps") ("wn" ("wa" "ph" ("wn" "ps")))) ("bitri" ("wi" "ph" "ps") ("wi" "ph" ("wn" ("wn" "ps"))) ("wn" ("wa" "ph" ("wn" "ps"))) ("imbi2i" "ps" ("wn" ("wn" "ps")) "ph" ("notnotb" "ps")) ("imnan" "ph" ("wn" "ps")))) ;; Express conjunction in terms of implication. (Contributed by NM, 2-Aug-1994.) (theorem "annim" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wn" "ps")) ("wn" ("wi" "ph" "ps"))) ("con2bii" ("wi" "ph" "ps") ("wa" "ph" ("wn" "ps")) ("iman" "ph" "ps"))) ;; Theorem *4.61 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_61" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wi" "ph" "ps")) ("wa" "ph" ("wn" "ps"))) ("bicomi" ("wa" "ph" ("wn" "ps")) ("wn" ("wi" "ph" "ps")) ("annim" "ph" "ps"))) ;; Theorem *4.65 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_65" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wi" ("wn" "ph") "ps")) ("wa" ("wn" "ph") ("wn" "ps"))) ("pm4_61" ("wn" "ph") "ps")) ;; Theorem *4.67 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_67" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wi" ("wn" "ph") ("wn" "ps"))) ("wa" ("wn" "ph") "ps")) ("pm4_63" ("wn" "ph") "ps")) ;; Importation inference. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) (theorem "imp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("imp_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" ("wa" "ph" "ps") "ch") ("sylbi" ("wa" "ph" "ps") ("wn" ("wi" "ph" ("wn" "ps"))) "ch" ("df_an" "ph" "ps") ("impi" "ph" "ps" "ch" "imp_1"))) ;; Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.) (theorem "impcom" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("imp_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" ("wa" "ps" "ph") "ch") ("imp" "ps" "ph" "ch" ("com12" "ph" "ps" "ch" "imp_1"))) ;; Importation deduction. (Contributed by NM, 31-Mar-1994.) (theorem "impd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("impd_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")) ("com12" ("wa" "ps" "ch") "ph" "th" ("imp" "ps" "ch" ("wi" "ph" "th") ("com3l" "ph" "ps" "ch" "th" "impd_1")))) ;; An importation inference. (Contributed by NM, 26-Apr-1994.) (theorem "imp31" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("impd_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" ("wa" ("wa" "ph" "ps") "ch") "th") ("imp" ("wa" "ph" "ps") "ch" "th" ("imp" "ph" "ps" ("wi" "ch" "th") "impd_1"))) ;; An importation inference. (Contributed by NM, 26-Apr-1994.) (theorem "imp32" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("impd_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th") ("imp" "ph" ("wa" "ps" "ch") "th" ("impd" "ph" "ps" "ch" "th" "impd_1"))) ;; Exportation inference. (This theorem used to be labeled "exp" but was changed to "ex" so as not to conflict with the math token "exp", per the June 2006 Metamath spec change.) A translation of natural deduction rule ` -> ` I ( ` -> ` introduction), see ~ natded . (Contributed by NM, 3-Jan-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) (theorem "ex" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ex_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" "ph" ("wi" "ps" "ch")) ("expi" "ph" "ps" "ch" ("sylbir" ("wn" ("wi" "ph" ("wn" "ps"))) ("wa" "ph" "ps") "ch" ("df_an" "ph" "ps") "ex_1"))) ;; Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.) (theorem "expcom" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ex_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" "ps" ("wi" "ph" "ch")) ("com12" "ph" "ps" "ch" ("ex" "ph" "ps" "ch" "ex_1"))) ;; Exportation deduction. (Contributed by NM, 20-Aug-1993.) (theorem "expd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("expd_1" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))) ("com3r" "ps" "ch" "ph" "th" ("ex" "ps" "ch" ("wi" "ph" "th") ("com12" "ph" ("wa" "ps" "ch") "th" "expd_1")))) ;; A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.) (theorem "expdimp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("expd_1" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" ("wa" "ph" "ps") ("wi" "ch" "th")) ("imp" "ph" "ps" ("wi" "ch" "th") ("expd" "ph" "ps" "ch" "th" "expd_1"))) ;; Deduction form of ~ expcom . (Contributed by Alan Sare, 22-Jul-2012.) (theorem "expcomd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("expcomd_1" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" ("wi" "ch" ("wi" "ps" "th"))) ("com23" "ph" "ps" "ch" "th" ("expd" "ph" "ps" "ch" "th" "expcomd_1"))) ;; Commuted form of ~ expd . (Contributed by Alan Sare, 18-Mar-2012.) (theorem "expdcom" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("expdcom_1" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" "ps" ("wi" "ch" ("wi" "ph" "th"))) ("com3l" "ph" "ps" "ch" "th" ("expd" "ph" "ps" "ch" "th" "expdcom_1"))) ;; Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.) (theorem "impancom" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("impancom_1" ("wi" ("wa" "ph" "ps") ("wi" "ch" "th")))) (for) ("wi" ("wa" "ph" "ch") ("wi" "ps" "th")) ("imp" "ph" "ch" ("wi" "ps" "th") ("com23" "ph" "ps" "ch" "th" ("ex" "ph" "ps" ("wi" "ch" "th") "impancom_1")))) ;; Variant of ~ con3d with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (theorem "con3dimp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("con3dimp_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" ("wa" "ph" ("wn" "ch")) ("wn" "ps")) ("imp" "ph" ("wn" "ch") ("wn" "ps") ("con3d" "ph" "ps" "ch" "con3dimp_1"))) ;; Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.) (theorem "pm2_01da" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm2_01da_1" ("wi" ("wa" "ph" "ps") ("wn" "ps")))) (for) ("wi" "ph" ("wn" "ps")) ("pm2_01d" "ph" "ps" ("ex" "ph" "ps" ("wn" "ps") "pm2_01da_1"))) ;; Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.) (theorem "pm2_18da" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm2_18da_1" ("wi" ("wa" "ph" ("wn" "ps")) "ps"))) (for) ("wi" "ph" "ps") ("pm2_18d" "ph" "ps" ("ex" "ph" ("wn" "ps") "ps" "pm2_18da_1"))) ;; Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) (theorem "pm3_3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" ("wa" "ph" "ps") "ch") ("wi" "ph" ("wi" "ps" "ch"))) ("expd" ("wi" ("wa" "ph" "ps") "ch") "ph" "ps" "ch" ("id" ("wi" ("wa" "ph" "ps") "ch")))) ;; Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) (theorem "pm3_31" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" ("wa" "ph" "ps") "ch")) ("impd" ("wi" "ph" ("wi" "ps" "ch")) "ph" "ps" "ch" ("id" ("wi" "ph" ("wi" "ps" "ch"))))) ;; Import-export theorem. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) (theorem "impexp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" ("wa" "ph" "ps") "ch") ("wi" "ph" ("wi" "ps" "ch"))) ("impbii" ("wi" ("wa" "ph" "ps") "ch") ("wi" "ph" ("wi" "ps" "ch")) ("pm3_3" "ph" "ps" "ch") ("pm3_31" "ph" "ps" "ch"))) ;; Join antecedents with conjunction ("conjunction introduction"). Theorem *3.2 of [WhiteheadRussell] p. 111. See ~ pm3.2im for a version using only implication and negation. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) (theorem "pm3_2" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wi" "ps" ("wa" "ph" "ps"))) ("ex" "ph" "ps" ("wa" "ph" "ps") ("id" ("wa" "ph" "ps")))) ;; Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.) (theorem "pm3_21" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wi" "ps" ("wa" "ps" "ph"))) ("com12" "ps" "ph" ("wa" "ps" "ph") ("pm3_2" "ps" "ph"))) ;; Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) (theorem "pm3_22" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") ("wa" "ps" "ph")) ("imp" "ph" "ps" ("wa" "ps" "ph") ("pm3_21" "ph" "ps"))) ;; Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Wolf Lammen, 4-Nov-2012.) (theorem "ancom" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" "ph" "ps") ("wa" "ps" "ph")) ("impbii" ("wa" "ph" "ps") ("wa" "ps" "ph") ("pm3_22" "ph" "ps") ("pm3_22" "ps" "ph"))) ;; Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.) (theorem "ancomd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ancomd_1" ("wi" "ph" ("wa" "ps" "ch")))) (for) ("wi" "ph" ("wa" "ch" "ps")) ("sylib" "ph" ("wa" "ps" "ch") ("wa" "ch" "ps") "ancomd_1" ("ancom" "ps" "ch"))) ;; Closed form of ~ ancoms . (Contributed by Alan Sare, 31-Dec-2011.) (theorem "ancomst" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" ("wa" "ph" "ps") "ch") ("wi" ("wa" "ps" "ph") "ch")) ("imbi1i" ("wa" "ph" "ps") ("wa" "ps" "ph") "ch" ("ancom" "ph" "ps"))) ;; Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.) (theorem "ancoms" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ancoms_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" "ps" "ph") "ch") ("imp" "ps" "ph" "ch" ("expcom" "ph" "ps" "ch" "ancoms_1"))) ;; Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.) (theorem "ancomsd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("ancomsd_1" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" ("wi" ("wa" "ch" "ps") "th")) ("syl5bi" ("wa" "ch" "ps") ("wa" "ps" "ch") "ph" "th" ("ancom" "ch" "ps") "ancomsd_1")) ;; Infer conjunction of premises. (Contributed by NM, 21-Jun-1993.) (theorem "pm3_2i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm3_2i_1" "ph") ("pm3_2i_2" "ps")) (for) ("wa" "ph" "ps") ("mp2" "ph" "ps" ("wa" "ph" "ps") "pm3_2i_1" "pm3_2i_2" ("pm3_2" "ph" "ps"))) ;; Nested conjunction of antecedents. (Contributed by NM, 4-Jan-1993.) (theorem "pm3_43i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wi" "ph" "ch") ("wi" "ph" ("wa" "ps" "ch")))) ("imim3i" "ps" "ch" ("wa" "ps" "ch") "ph" ("pm3_2" "ps" "ch"))) ;; Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.) (theorem "adantr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("adantr_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" "ph" "ch") "ps") ("imp" "ph" "ch" "ps" ("a1d" "ph" "ps" "ch" "adantr_1"))) ;; Inference adding a conjunct to the left of an antecedent. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) (theorem "adantl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("adantl_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" "ch" "ph") "ps") ("ancoms" "ph" "ch" "ps" ("adantr" "ph" "ps" "ch" "adantl_1"))) ;; Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) (Proof shortened by Wolf Lammen, 14-Jun-2022.) (theorem "simpl" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") "ph") ("adantr" "ph" "ph" "ps" ("id" "ph"))) ;; Obsolete proof of ~ simpl as of 14-Jun-2022. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simplOLD" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") "ph") ("imp" "ph" "ps" "ph" ("ax_1" "ph" "ps"))) ;; Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) (theorem "simpli" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("simpli_1" ("wa" "ph" "ps"))) (for) "ph" ("ax_mp" ("wa" "ph" "ps") "ph" "simpli_1" ("simpl" "ph" "ps"))) ;; Deduction eliminating a conjunct. A translation of natural deduction rule ` /\ ` EL ( ` /\ ` elimination left), see ~ natded . (Contributed by NM, 26-May-1993.) (theorem "simpld" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("simpld_1" ("wi" "ph" ("wa" "ps" "ch")))) (for) ("wi" "ph" "ps") ("syl" "ph" ("wa" "ps" "ch") "ps" "simpld_1" ("simpl" "ps" "ch"))) ;; Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.) (theorem "simplbi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("simplbi_1" ("wb" "ph" ("wa" "ps" "ch")))) (for) ("wi" "ph" "ps") ("simpld" "ph" "ps" "ch" ("biimpi" "ph" ("wa" "ps" "ch") "simplbi_1"))) ;; Elimination of a conjunct. Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) (Proof shortened by Wolf Lammen, 14-Jun-2022.) (theorem "simpr" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") "ps") ("adantl" "ps" "ps" "ph" ("id" "ps"))) ;; Obsolete proof of ~ simpr as of 14-Jun-2022. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simprOLD" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") "ps") ("imp" "ph" "ps" "ps" ("idd" "ph" "ps"))) ;; Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) (theorem "simpri" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("simpri_1" ("wa" "ph" "ps"))) (for) "ps" ("ax_mp" ("wa" "ph" "ps") "ps" "simpri_1" ("simpr" "ph" "ps"))) ;; Deduction eliminating a conjunct. (Contributed by NM, 14-May-1993.) A translation of natural deduction rule ` /\ ` ER ( ` /\ ` elimination right), see ~ natded . (Proof shortened by Wolf Lammen, 3-Oct-2013.) (theorem "simprd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("simprd_1" ("wi" "ph" ("wa" "ps" "ch")))) (for) ("wi" "ph" "ch") ("simpld" "ph" "ch" "ps" ("ancomd" "ph" "ps" "ch" "simprd_1"))) ;; Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.) (theorem "simprbi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("simprbi_1" ("wb" "ph" ("wa" "ps" "ch")))) (for) ("wi" "ph" "ch") ("simprd" "ph" "ps" "ch" ("biimpi" "ph" ("wa" "ps" "ch") "simprbi_1"))) ;; Deduction adding a conjunct to the left of an antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.) (theorem "adantld" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("adantld_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" ("wa" "th" "ps") "ch")) ("syl5" ("wa" "th" "ps") "ps" "ph" "ch" ("simpr" "th" "ps") "adantld_1")) ;; Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.) (theorem "adantrd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("adantrd_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" ("wa" "ps" "th") "ch")) ("syl5" ("wa" "ps" "th") "ps" "ph" "ch" ("simpl" "ps" "th") "adantrd_1")) ;; An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.) (theorem "impel" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("impel_1" ("wi" "ph" ("wi" "ps" "ch"))) ("impel_2" ("wi" "th" "ps"))) (for) ("wi" ("wa" "ph" "th") "ch") ("imp" "ph" "th" "ch" ("syl5" "th" "ps" "ph" "ch" "impel_2" "impel_1"))) ;; Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "mpan9" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpan9_1" ("wi" "ph" "ps")) ("mpan9_2" ("wi" "ch" ("wi" "ps" "th")))) (for) ("wi" ("wa" "ph" "ch") "th") ("impcom" "ch" "ph" "th" ("syl5" "ph" "ps" "ch" "th" "mpan9_1" "mpan9_2"))) ;; A syllogism deduction with conjoined antecedents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.) (theorem "syldan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syldan_1" ("wi" ("wa" "ph" "ps") "ch")) ("syldan_2" ("wi" ("wa" "ph" "ch") "th"))) (for) ("wi" ("wa" "ph" "ps") "th") ("mpcom" "ch" ("wa" "ph" "ps") "th" "syldan_1" ("adantrd" "ch" "ph" "th" "ps" ("expcom" "ph" "ch" "th" "syldan_2")))) ;; A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.) (theorem "sylan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylan_1" ("wi" "ph" "ps")) ("sylan_2" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" "ch") "th") ("mpan9" "ph" "ps" "ch" "th" "sylan_1" ("expcom" "ps" "ch" "th" "sylan_2"))) ;; A syllogism inference. (Contributed by NM, 18-May-1994.) (theorem "sylanb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylanb_1" ("wb" "ph" "ps")) ("sylanb_2" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" "ch") "th") ("sylan" "ph" "ps" "ch" "th" ("biimpi" "ph" "ps" "sylanb_1") "sylanb_2")) ;; A syllogism inference. (Contributed by NM, 18-May-1994.) (theorem "sylanbr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylanbr_1" ("wb" "ps" "ph")) ("sylanbr_2" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" "ch") "th") ("sylan" "ph" "ps" "ch" "th" ("biimpri" "ps" "ph" "sylanbr_1") "sylanbr_2")) ;; A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.) (theorem "sylan2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylan2_1" ("wi" "ph" "ch")) ("sylan2_2" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" ("wa" "ps" "ph") "th") ("syldan" "ps" "ph" "ch" "th" ("adantl" "ph" "ch" "ps" "sylan2_1") "sylan2_2")) ;; A syllogism inference. (Contributed by NM, 21-Apr-1994.) (theorem "sylan2b" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylan2b_1" ("wb" "ph" "ch")) ("sylan2b_2" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" ("wa" "ps" "ph") "th") ("sylan2" "ph" "ps" "ch" "th" ("biimpi" "ph" "ch" "sylan2b_1") "sylan2b_2")) ;; A syllogism inference. (Contributed by NM, 21-Apr-1994.) (theorem "sylan2br" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylan2br_1" ("wb" "ch" "ph")) ("sylan2br_2" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" ("wa" "ps" "ph") "th") ("sylan2" "ph" "ps" "ch" "th" ("biimpri" "ch" "ph" "sylan2br_1") "sylan2br_2")) ;; A double syllogism inference. For an implication-only version, see ~ syl2im . (Contributed by NM, 31-Jan-1997.) (theorem "syl2an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl2an_1" ("wi" "ph" "ps")) ("syl2an_2" ("wi" "ta" "ch")) ("syl2an_3" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" "ta") "th") ("sylan2" "ta" "ph" "ch" "th" "syl2an_2" ("sylan" "ph" "ps" "ch" "th" "syl2an_1" "syl2an_3"))) ;; A double syllogism inference. For an implication-only version, see ~ syl2imc . (Contributed by NM, 17-Sep-2013.) (theorem "syl2anr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl2an_1" ("wi" "ph" "ps")) ("syl2an_2" ("wi" "ta" "ch")) ("syl2an_3" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" ("wa" "ta" "ph") "th") ("ancoms" "ph" "ta" "th" ("syl2an" "ph" "ps" "ch" "th" "ta" "syl2an_1" "syl2an_2" "syl2an_3"))) ;; A double syllogism inference. (Contributed by NM, 29-Jul-1999.) (theorem "syl2anb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl2anb_1" ("wb" "ph" "ps")) ("syl2anb_2" ("wb" "ta" "ch")) ("syl2anb_3" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" "ta") "th") ("sylan2b" "ta" "ph" "ch" "th" "syl2anb_2" ("sylanb" "ph" "ps" "ch" "th" "syl2anb_1" "syl2anb_3"))) ;; A double syllogism inference. (Contributed by NM, 29-Jul-1999.) (theorem "syl2anbr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl2anbr_1" ("wb" "ps" "ph")) ("syl2anbr_2" ("wb" "ch" "ta")) ("syl2anbr_3" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" "ta") "th") ("sylan2br" "ta" "ph" "ch" "th" "syl2anbr_2" ("sylanbr" "ph" "ps" "ch" "th" "syl2anbr_1" "syl2anbr_3"))) ;; A syllogism deduction. (Contributed by NM, 15-Dec-2004.) (theorem "syland" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syland_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syland_2" ("wi" "ph" ("wi" ("wa" "ch" "th") "ta")))) (for) ("wi" "ph" ("wi" ("wa" "ps" "th") "ta")) ("impd" "ph" "ps" "th" "ta" ("syld" "ph" "ps" "ch" ("wi" "th" "ta") "syland_1" ("expd" "ph" "ch" "th" "ta" "syland_2")))) ;; A syllogism deduction. (Contributed by NM, 15-Dec-2004.) (theorem "sylan2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylan2d_1" ("wi" "ph" ("wi" "ps" "ch"))) ("sylan2d_2" ("wi" "ph" ("wi" ("wa" "th" "ch") "ta")))) (for) ("wi" "ph" ("wi" ("wa" "th" "ps") "ta")) ("ancomsd" "ph" "ps" "th" "ta" ("syland" "ph" "ps" "ch" "th" "ta" "sylan2d_1" ("ancomsd" "ph" "th" "ch" "ta" "sylan2d_2")))) ;; A syllogism deduction. (Contributed by NM, 15-Dec-2004.) (theorem "syl2and" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl2and_1" ("wi" "ph" ("wi" "ps" "ch"))) ("syl2and_2" ("wi" "ph" ("wi" "th" "ta"))) ("syl2and_3" ("wi" "ph" ("wi" ("wa" "ch" "ta") "et")))) (for) ("wi" "ph" ("wi" ("wa" "ps" "th") "et")) ("syland" "ph" "ps" "ch" "th" "et" "syl2and_1" ("sylan2d" "ph" "th" "ta" "ch" "et" "syl2and_2" "syl2and_3"))) ;; Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.) (theorem "biimpa" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("biimpa_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" ("wa" "ph" "ps") "ch") ("imp" "ph" "ps" "ch" ("biimpd" "ph" "ps" "ch" "biimpa_1"))) ;; Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.) (theorem "biimpar" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("biimpa_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" ("wa" "ph" "ch") "ps") ("imp" "ph" "ch" "ps" ("biimprd" "ph" "ps" "ch" "biimpa_1"))) ;; Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.) (theorem "biimpac" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("biimpa_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" ("wa" "ps" "ph") "ch") ("imp" "ps" "ph" "ch" ("biimpcd" "ph" "ps" "ch" "biimpa_1"))) ;; Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.) (theorem "biimparc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("biimpa_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" ("wa" "ch" "ph") "ps") ("imp" "ch" "ph" "ps" ("biimprcd" "ph" "ps" "ch" "biimpa_1"))) ;; Conjunction implies disjunction with one common formula (1/4). (Contributed by BJ, 4-Oct-2019.) (theorem "animorl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") ("wo" "ph" "ch")) ("orcd" ("wa" "ph" "ps") "ph" "ch" ("simpl" "ph" "ps"))) ;; Conjunction implies disjunction with one common formula (2/4). (Contributed by BJ, 4-Oct-2019.) (theorem "animorr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") ("wo" "ch" "ps")) ("olcd" ("wa" "ph" "ps") "ps" "ch" ("simpr" "ph" "ps"))) ;; Conjunction implies disjunction with one common formula (3/4). (Contributed by BJ, 4-Oct-2019.) (theorem "animorlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") ("wo" "ch" "ph")) ("olcd" ("wa" "ph" "ps") "ph" "ch" ("simpl" "ph" "ps"))) ;; Conjunction implies disjunction with one common formula (4/4). (Contributed by BJ, 4-Oct-2019.) (theorem "animorrl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") ("wo" "ps" "ch")) ("orcd" ("wa" "ph" "ps") "ps" "ch" ("simpr" "ph" "ps"))) ;; Negated conjunction in terms of disjunction (De Morgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "ianor" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wa" "ph" "ps")) ("wo" ("wn" "ph") ("wn" "ps"))) ("bitr3i" ("wn" ("wa" "ph" "ps")) ("wi" "ph" ("wn" "ps")) ("wo" ("wn" "ph") ("wn" "ps")) ("imnan" "ph" "ps") ("pm4_62" "ph" "ps"))) ;; Conjunction in terms of disjunction (De Morgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 3-Nov-2012.) (theorem "anor" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" "ph" "ps") ("wn" ("wo" ("wn" "ph") ("wn" "ps")))) ("con2bii" ("wo" ("wn" "ph") ("wn" "ps")) ("wa" "ph" "ps") ("bicomi" ("wn" ("wa" "ph" "ps")) ("wo" ("wn" "ph") ("wn" "ps")) ("ianor" "ph" "ps")))) ;; Negated disjunction in terms of conjunction (De Morgan's law). Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "ioran" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wo" "ph" "ps")) ("wa" ("wn" "ph") ("wn" "ps"))) ("xchnxbi" ("wi" ("wn" "ph") "ps") ("wa" ("wn" "ph") ("wn" "ps")) ("wo" "ph" "ps") ("pm4_65" "ph" "ps") ("pm4_64" "ph" "ps"))) ;; Theorem *4.52 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.) (theorem "pm4_52" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wn" "ps")) ("wn" ("wo" ("wn" "ph") "ps"))) ("xchbinx" ("wa" "ph" ("wn" "ps")) ("wi" "ph" "ps") ("wo" ("wn" "ph") "ps") ("annim" "ph" "ps") ("imor" "ph" "ps"))) ;; Theorem *4.53 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_53" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wa" "ph" ("wn" "ps"))) ("wo" ("wn" "ph") "ps")) ("bicomi" ("wo" ("wn" "ph") "ps") ("wn" ("wa" "ph" ("wn" "ps"))) ("con2bii" ("wa" "ph" ("wn" "ps")) ("wo" ("wn" "ph") "ps") ("pm4_52" "ph" "ps")))) ;; Theorem *4.54 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.) (theorem "pm4_54" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" ("wn" "ph") "ps") ("wn" ("wo" "ph" ("wn" "ps")))) ("xchbinx" ("wa" ("wn" "ph") "ps") ("wi" ("wn" "ph") ("wn" "ps")) ("wo" "ph" ("wn" "ps")) ("df_an" ("wn" "ph") "ps") ("pm4_66" "ph" "ps"))) ;; Theorem *4.55 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_55" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wa" ("wn" "ph") "ps")) ("wo" "ph" ("wn" "ps"))) ("bicomi" ("wo" "ph" ("wn" "ps")) ("wn" ("wa" ("wn" "ph") "ps")) ("con2bii" ("wa" ("wn" "ph") "ps") ("wo" "ph" ("wn" "ps")) ("pm4_54" "ph" "ps")))) ;; Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_56" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" ("wn" "ph") ("wn" "ps")) ("wn" ("wo" "ph" "ps"))) ("bicomi" ("wn" ("wo" "ph" "ps")) ("wa" ("wn" "ph") ("wn" "ps")) ("ioran" "ph" "ps"))) ;; Disjunction in terms of conjunction (De Morgan's law). Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "oran" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wo" "ph" "ps") ("wn" ("wa" ("wn" "ph") ("wn" "ps")))) ("con2bii" ("wa" ("wn" "ph") ("wn" "ps")) ("wo" "ph" "ps") ("pm4_56" "ph" "ps"))) ;; Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_57" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wa" ("wn" "ph") ("wn" "ps"))) ("wo" "ph" "ps")) ("bicomi" ("wo" "ph" "ps") ("wn" ("wa" ("wn" "ph") ("wn" "ps"))) ("oran" "ph" "ps"))) ;; Theorem *3.1 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (theorem "pm3_1" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") ("wn" ("wo" ("wn" "ph") ("wn" "ps")))) ("biimpi" ("wa" "ph" "ps") ("wn" ("wo" ("wn" "ph") ("wn" "ps"))) ("anor" "ph" "ps"))) ;; Theorem *3.11 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (theorem "pm3_11" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wo" ("wn" "ph") ("wn" "ps"))) ("wa" "ph" "ps")) ("biimpri" ("wa" "ph" "ps") ("wn" ("wo" ("wn" "ph") ("wn" "ps"))) ("anor" "ph" "ps"))) ;; Theorem *3.12 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (theorem "pm3_12" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wo" ("wo" ("wn" "ph") ("wn" "ps")) ("wa" "ph" "ps")) ("orri" ("wo" ("wn" "ph") ("wn" "ps")) ("wa" "ph" "ps") ("pm3_11" "ph" "ps"))) ;; Theorem *3.13 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (theorem "pm3_13" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wa" "ph" "ps")) ("wo" ("wn" "ph") ("wn" "ps"))) ("con1i" ("wo" ("wn" "ph") ("wn" "ps")) ("wa" "ph" "ps") ("pm3_11" "ph" "ps"))) ;; Theorem *3.14 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (theorem "pm3_14" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wo" ("wn" "ph") ("wn" "ps")) ("wn" ("wa" "ph" "ps"))) ("con2i" ("wa" "ph" "ps") ("wo" ("wn" "ph") ("wn" "ps")) ("pm3_1" "ph" "ps"))) ;; Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) (theorem "iba" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wb" "ps" ("wa" "ps" "ph"))) ("impbid1" "ph" "ps" ("wa" "ps" "ph") ("pm3_21" "ph" "ps") ("simpl" "ps" "ph"))) ;; Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.) (theorem "ibar" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wb" "ps" ("wa" "ph" "ps"))) ("impbid1" "ph" "ps" ("wa" "ph" "ps") ("pm3_2" "ph" "ps") ("simpr" "ph" "ps"))) ;; A wff is equivalent to its conjunction with truth. (Contributed by NM, 26-May-1993.) (theorem "biantru" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("biantru_1" "ph")) (for) ("wb" "ps" ("wa" "ps" "ph")) ("ax_mp" "ph" ("wb" "ps" ("wa" "ps" "ph")) "biantru_1" ("iba" "ph" "ps"))) ;; A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.) (theorem "biantrur" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("biantrur_1" "ph")) (for) ("wb" "ps" ("wa" "ph" "ps")) ("ax_mp" "ph" ("wb" "ps" ("wa" "ph" "ps")) "biantrur_1" ("ibar" "ph" "ps"))) ;; A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.) (theorem "biantrud" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("biantrud_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("wb" "ch" ("wa" "ch" "ps"))) ("syl" "ph" "ps" ("wb" "ch" ("wa" "ch" "ps")) "biantrud_1" ("iba" "ps" "ch"))) ;; A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "biantrurd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("biantrud_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("wb" "ch" ("wa" "ps" "ch"))) ("syl" "ph" "ps" ("wb" "ch" ("wa" "ps" "ch")) "biantrud_1" ("ibar" "ps" "ch"))) ;; Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (theorem "mpbirand" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpbirand_1" ("wi" "ph" "ch")) ("mpbirand_2" ("wi" "ph" ("wb" "ps" ("wa" "ch" "th"))))) (for) ("wi" "ph" ("wb" "ps" "th")) ("bitr4d" "ph" "ps" ("wa" "ch" "th") "th" "mpbirand_2" ("biantrurd" "ph" "ch" "th" "mpbirand_1"))) ;; Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.) (theorem "jaao" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("jaao_1" ("wi" "ph" ("wi" "ps" "ch"))) ("jaao_2" ("wi" "th" ("wi" "ta" "ch")))) (for) ("wi" ("wa" "ph" "th") ("wi" ("wo" "ps" "ta") "ch")) ("jaod" ("wa" "ph" "th") "ps" "ch" "ta" ("adantr" "ph" ("wi" "ps" "ch") "th" "jaao_1") ("adantl" "th" ("wi" "ta" "ch") "ph" "jaao_2"))) ;; Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.) (theorem "jaoa" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("jaao_1" ("wi" "ph" ("wi" "ps" "ch"))) ("jaao_2" ("wi" "th" ("wi" "ta" "ch")))) (for) ("wi" ("wo" "ph" "th") ("wi" ("wa" "ps" "ta") "ch")) ("jaoi" "ph" ("wi" ("wa" "ps" "ta") "ch") "th" ("adantrd" "ph" "ps" "ch" "ta" "jaao_1") ("adantld" "th" "ta" "ch" "ps" "jaao_2"))) ;; Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) (theorem "pm3_44" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" ("wi" "ps" "ph") ("wi" "ch" "ph")) ("wi" ("wo" "ps" "ch") "ph")) ("jaao" ("wi" "ps" "ph") "ps" "ph" ("wi" "ch" "ph") "ch" ("id" ("wi" "ps" "ph")) ("id" ("wi" "ch" "ph")))) ;; Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 4-Apr-2013.) (theorem "jao" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wi" "ch" "ps") ("wi" ("wo" "ph" "ch") "ps"))) ("ex" ("wi" "ph" "ps") ("wi" "ch" "ps") ("wi" ("wo" "ph" "ch") "ps") ("pm3_44" "ps" "ph" "ch"))) ;; Axiom *1.2 of [WhiteheadRussell] p. 96, which they call "Taut". (Contributed by NM, 3-Jan-2005.) (theorem "pm1_2" (for ("ph" ( "wff"))) (for) (for) ("wi" ("wo" "ph" "ph") "ph") ("jaoi" "ph" "ph" "ph" ("id" "ph") ("id" "ph"))) ;; Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.) (Proof shortened by Andrew Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.) (theorem "oridm" (for ("ph" ( "wff"))) (for) (for) ("wb" ("wo" "ph" "ph") "ph") ("impbii" ("wo" "ph" "ph") "ph" ("pm1_2" "ph") ("pm2_07" "ph"))) ;; Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_25" (for ("ph" ( "wff"))) (for) (for) ("wb" "ph" ("wo" "ph" "ph")) ("bicomi" ("wo" "ph" "ph") "ph" ("oridm" "ph"))) ;; Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.) (theorem "orim12i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("orim12i_1" ("wi" "ph" "ps")) ("orim12i_2" ("wi" "ch" "th"))) (for) ("wi" ("wo" "ph" "ch") ("wo" "ps" "th")) ("jaoi" "ph" ("wo" "ps" "th") "ch" ("orcd" "ph" "ps" "th" "orim12i_1") ("olcd" "ch" "th" "ps" "orim12i_2"))) ;; Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.) (theorem "orim1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("orim1i_1" ("wi" "ph" "ps"))) (for) ("wi" ("wo" "ph" "ch") ("wo" "ps" "ch")) ("orim12i" "ph" "ps" "ch" "ch" "orim1i_1" ("id" "ch"))) ;; Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.) (theorem "orim2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("orim1i_1" ("wi" "ph" "ps"))) (for) ("wi" ("wo" "ch" "ph") ("wo" "ch" "ps")) ("orim12i" "ch" "ch" "ph" "ps" ("id" "ch") "orim1i_1")) ;; Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Dec-2012.) (theorem "orbi2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("orbi2i_1" ("wb" "ph" "ps"))) (for) ("wb" ("wo" "ch" "ph") ("wo" "ch" "ps")) ("impbii" ("wo" "ch" "ph") ("wo" "ch" "ps") ("orim2i" "ph" "ps" "ch" ("biimpi" "ph" "ps" "orbi2i_1")) ("orim2i" "ps" "ph" "ch" ("biimpri" "ph" "ps" "orbi2i_1")))) ;; Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (theorem "orbi1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("orbi2i_1" ("wb" "ph" "ps"))) (for) ("wb" ("wo" "ph" "ch") ("wo" "ps" "ch")) ("_3bitri" ("wo" "ph" "ch") ("wo" "ch" "ph") ("wo" "ch" "ps") ("wo" "ps" "ch") ("orcom" "ph" "ch") ("orbi2i" "ph" "ps" "ch" "orbi2i_1") ("orcom" "ch" "ps"))) ;; Infer the disjunction of two equivalences. (Contributed by NM, 3-Jan-1993.) (theorem "orbi12i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("orbi12i_1" ("wb" "ph" "ps")) ("orbi12i_2" ("wb" "ch" "th"))) (for) ("wb" ("wo" "ph" "ch") ("wo" "ps" "th")) ("bitri" ("wo" "ph" "ch") ("wo" "ph" "th") ("wo" "ps" "th") ("orbi2i" "ch" "th" "ph" "orbi12i_2") ("orbi1i" "ph" "ps" "th" "orbi12i_1"))) ;; Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (theorem "pm1_5" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wo" "ph" ("wo" "ps" "ch")) ("wo" "ps" ("wo" "ph" "ch"))) ("jaoi" "ph" ("wo" "ps" ("wo" "ph" "ch")) ("wo" "ps" "ch") ("olcd" "ph" ("wo" "ph" "ch") "ps" ("orc" "ph" "ch")) ("orim2i" "ch" ("wo" "ph" "ch") "ps" ("olc" "ch" "ph")))) ;; Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) (theorem "or12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" "ph" ("wo" "ps" "ch")) ("wo" "ps" ("wo" "ph" "ch"))) ("impbii" ("wo" "ph" ("wo" "ps" "ch")) ("wo" "ps" ("wo" "ph" "ch")) ("pm1_5" "ph" "ps" "ch") ("pm1_5" "ps" "ph" "ch"))) ;; Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (theorem "orass" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" ("wo" "ph" "ps") "ch") ("wo" "ph" ("wo" "ps" "ch"))) ("_3bitri" ("wo" ("wo" "ph" "ps") "ch") ("wo" "ch" ("wo" "ph" "ps")) ("wo" "ph" ("wo" "ch" "ps")) ("wo" "ph" ("wo" "ps" "ch")) ("orcom" ("wo" "ph" "ps") "ch") ("or12" "ch" "ph" "ps") ("orbi2i" ("wo" "ch" "ps") ("wo" "ps" "ch") "ph" ("orcom" "ch" "ps")))) ;; Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_31" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wo" "ph" ("wo" "ps" "ch")) ("wo" ("wo" "ph" "ps") "ch")) ("biimpri" ("wo" ("wo" "ph" "ps") "ch") ("wo" "ph" ("wo" "ps" "ch")) ("orass" "ph" "ps" "ch"))) ;; Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_32" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wo" ("wo" "ph" "ps") "ch") ("wo" "ph" ("wo" "ps" "ch"))) ("biimpi" ("wo" ("wo" "ph" "ps") "ch") ("wo" "ph" ("wo" "ps" "ch")) ("orass" "ph" "ps" "ch"))) ;; A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (theorem "or32" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" ("wo" "ph" "ps") "ch") ("wo" ("wo" "ph" "ch") "ps")) ("_3bitri" ("wo" ("wo" "ph" "ps") "ch") ("wo" "ph" ("wo" "ps" "ch")) ("wo" "ps" ("wo" "ph" "ch")) ("wo" ("wo" "ph" "ch") "ps") ("orass" "ph" "ps" "ch") ("or12" "ph" "ps" "ch") ("orcom" "ps" ("wo" "ph" "ch")))) ;; Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.) (theorem "or4" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wo" ("wo" "ph" "ps") ("wo" "ch" "th")) ("wo" ("wo" "ph" "ch") ("wo" "ps" "th"))) ("_3bitr4i" ("wo" "ph" ("wo" "ps" ("wo" "ch" "th"))) ("wo" "ph" ("wo" "ch" ("wo" "ps" "th"))) ("wo" ("wo" "ph" "ps") ("wo" "ch" "th")) ("wo" ("wo" "ph" "ch") ("wo" "ps" "th")) ("orbi2i" ("wo" "ps" ("wo" "ch" "th")) ("wo" "ch" ("wo" "ps" "th")) "ph" ("or12" "ps" "ch" "th")) ("orass" "ph" "ps" ("wo" "ch" "th")) ("orass" "ph" "ch" ("wo" "ps" "th")))) ;; Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.) (theorem "or42" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wo" ("wo" "ph" "ps") ("wo" "ch" "th")) ("wo" ("wo" "ph" "ch") ("wo" "th" "ps"))) ("bitri" ("wo" ("wo" "ph" "ps") ("wo" "ch" "th")) ("wo" ("wo" "ph" "ch") ("wo" "ps" "th")) ("wo" ("wo" "ph" "ch") ("wo" "th" "ps")) ("or4" "ph" "ps" "ch" "th") ("orbi2i" ("wo" "ps" "th") ("wo" "th" "ps") ("wo" "ph" "ch") ("orcom" "ps" "th")))) ;; Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.) (theorem "orordi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" "ph" ("wo" "ps" "ch")) ("wo" ("wo" "ph" "ps") ("wo" "ph" "ch"))) ("bitr3i" ("wo" "ph" ("wo" "ps" "ch")) ("wo" ("wo" "ph" "ph") ("wo" "ps" "ch")) ("wo" ("wo" "ph" "ps") ("wo" "ph" "ch")) ("orbi1i" ("wo" "ph" "ph") "ph" ("wo" "ps" "ch") ("oridm" "ph")) ("or4" "ph" "ph" "ps" "ch"))) ;; Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.) (theorem "orordir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" ("wo" "ph" "ps") "ch") ("wo" ("wo" "ph" "ch") ("wo" "ps" "ch"))) ("bitr3i" ("wo" ("wo" "ph" "ps") "ch") ("wo" ("wo" "ph" "ps") ("wo" "ch" "ch")) ("wo" ("wo" "ph" "ch") ("wo" "ps" "ch")) ("orbi2i" ("wo" "ch" "ch") "ch" ("wo" "ph" "ps") ("oridm" "ch")) ("or4" "ph" "ps" "ch" "ch"))) ;; Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). Equivalent to the natural deduction rule ` /\ ` I ( ` /\ ` introduction), see ~ natded . (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) (theorem "jca" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("jca_1" ("wi" "ph" "ps")) ("jca_2" ("wi" "ph" "ch"))) (for) ("wi" "ph" ("wa" "ps" "ch")) ("sylc" "ph" "ps" "ch" ("wa" "ps" "ch") "jca_1" "jca_2" ("pm3_2" "ps" "ch"))) ;; Deduction conjoining the consequents of two implications. (Contributed by NM, 15-Jul-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.) (theorem "jcad" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jcad_1" ("wi" "ph" ("wi" "ps" "ch"))) ("jcad_2" ("wi" "ph" ("wi" "ps" "th")))) (for) ("wi" "ph" ("wi" "ps" ("wa" "ch" "th"))) ("syl6c" "ph" "ps" "ch" "th" ("wa" "ch" "th") "jcad_1" "jcad_2" ("pm3_2" "ch" "th"))) ;; Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.) (theorem "jca2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jca2_1" ("wi" "ph" ("wi" "ps" "ch"))) ("jca2_2" ("wi" "ps" "th"))) (for) ("wi" "ph" ("wi" "ps" ("wa" "ch" "th"))) ("jcad" "ph" "ps" "ch" "th" "jca2_1" ("a1i" ("wi" "ps" "th") "ph" "jca2_2"))) ;; Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.) (theorem "jca31" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jca31_1" ("wi" "ph" "ps")) ("jca31_2" ("wi" "ph" "ch")) ("jca31_3" ("wi" "ph" "th"))) (for) ("wi" "ph" ("wa" ("wa" "ps" "ch") "th")) ("jca" "ph" ("wa" "ps" "ch") "th" ("jca" "ph" "ps" "ch" "jca31_1" "jca31_2") "jca31_3")) ;; Join three consequents. (Contributed by FL, 1-Aug-2009.) (theorem "jca32" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jca31_1" ("wi" "ph" "ps")) ("jca31_2" ("wi" "ph" "ch")) ("jca31_3" ("wi" "ph" "th"))) (for) ("wi" "ph" ("wa" "ps" ("wa" "ch" "th"))) ("jca" "ph" "ps" ("wa" "ch" "th") "jca31_1" ("jca" "ph" "ch" "th" "jca31_2" "jca31_3"))) ;; Deduction replacing implication with conjunction. (Contributed by NM, 15-Jul-1993.) (theorem "jcai" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("jcai_1" ("wi" "ph" "ps")) ("jcai_2" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wa" "ps" "ch")) ("jca" "ph" "ps" "ch" "jcai_1" ("mpd" "ph" "ps" "ch" "jcai_1" "jcai_2"))) ;; Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.) (theorem "jctil" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("jctil_1" ("wi" "ph" "ps")) ("jctil_2" "ch")) (for) ("wi" "ph" ("wa" "ch" "ps")) ("jca" "ph" "ch" "ps" ("a1i" "ch" "ph" "jctil_2") "jctil_1")) ;; Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.) (theorem "jctir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("jctil_1" ("wi" "ph" "ps")) ("jctil_2" "ch")) (for) ("wi" "ph" ("wa" "ps" "ch")) ("jca" "ph" "ps" "ch" "jctil_1" ("a1i" "ch" "ph" "jctil_2"))) ;; Inference conjoining a consequent of a consequent to the right of the consequent in an implication. See also ~ ex-natded5.3i . (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by AV, 20-Aug-2019.) (theorem "jccir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("jccir_1" ("wi" "ph" "ps")) ("jccir_2" ("wi" "ps" "ch"))) (for) ("wi" "ph" ("wa" "ps" "ch")) ("jca" "ph" "ps" "ch" "jccir_1" ("syl" "ph" "ps" "ch" "jccir_1" "jccir_2"))) ;; Inference conjoining a consequent of a consequent to the left of the consequent in an implication. Remark: One can also prove this theorem using ~ syl and ~ jca (as done in ~ jccir ), which would be 4 bytes shorter, but one step longer than the current proof. (Proof modification is discouraged.) (Contributed by AV, 20-Aug-2019.) (theorem "jccil" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("jccir_1" ("wi" "ph" "ps")) ("jccir_2" ("wi" "ps" "ch"))) (for) ("wi" "ph" ("wa" "ch" "ps")) ("ancomd" "ph" "ps" "ch" ("jccir" "ph" "ps" "ch" "jccir_1" "jccir_2"))) ;; Inference conjoining a theorem to the left of a consequent. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.) (theorem "jctl" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("jctl_1" "ps")) (for) ("wi" "ph" ("wa" "ps" "ph")) ("jctil" "ph" "ph" "ps" ("id" "ph") "jctl_1")) ;; Inference conjoining a theorem to the right of a consequent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.) (theorem "jctr" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("jctl_1" "ps")) (for) ("wi" "ph" ("wa" "ph" "ps")) ("jctir" "ph" "ph" "ps" ("id" "ph") "jctl_1")) ;; Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.) (theorem "jctild" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jctild_1" ("wi" "ph" ("wi" "ps" "ch"))) ("jctild_2" ("wi" "ph" "th"))) (for) ("wi" "ph" ("wi" "ps" ("wa" "th" "ch"))) ("jcad" "ph" "ps" "th" "ch" ("a1d" "ph" "th" "ps" "jctild_2") "jctild_1")) ;; Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.) (theorem "jctird" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jctird_1" ("wi" "ph" ("wi" "ps" "ch"))) ("jctird_2" ("wi" "ph" "th"))) (for) ("wi" "ph" ("wi" "ps" ("wa" "ch" "th"))) ("jcad" "ph" "ps" "ch" "th" "jctird_1" ("a1d" "ph" "th" "ps" "jctird_2"))) ;; A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.) (theorem "syl6an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl6an_1" ("wi" "ph" "ps")) ("syl6an_2" ("wi" "ph" ("wi" "ch" "th"))) ("syl6an_3" ("wi" ("wa" "ps" "th") "ta"))) (for) ("wi" "ph" ("wi" "ch" "ta")) ("syl6" "ph" "ch" ("wa" "ps" "th") "ta" ("jctild" "ph" "ch" "th" "ps" "syl6an_2" "syl6an_1") "syl6an_3")) ;; Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.) (theorem "ancl" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" "ph" ("wa" "ph" "ps"))) ("a2i" "ph" "ps" ("wa" "ph" "ps") ("pm3_2" "ph" "ps"))) ;; Conjoin antecedent to left of consequent. Theorem *4.7 of [WhiteheadRussell] p. 120. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) (theorem "anclb" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" "ps") ("wi" "ph" ("wa" "ph" "ps"))) ("pm5_74i" "ph" "ps" ("wa" "ph" "ps") ("ibar" "ph" "ps"))) ;; Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_42" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ph" ("wi" "ps" ("wa" "ph" "ch")))) ("pm5_74i" "ph" ("wi" "ps" "ch") ("wi" "ps" ("wa" "ph" "ch")) ("imbi2d" "ph" "ch" ("wa" "ph" "ch") "ps" ("ibar" "ph" "ch")))) ;; Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.) (theorem "ancr" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" "ph" ("wa" "ps" "ph"))) ("a2i" "ph" "ps" ("wa" "ps" "ph") ("pm3_21" "ph" "ps"))) ;; Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) (theorem "ancrb" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" "ps") ("wi" "ph" ("wa" "ps" "ph"))) ("pm5_74i" "ph" "ps" ("wa" "ps" "ph") ("iba" "ph" "ps"))) ;; Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.) (theorem "ancli" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("ancli_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("wa" "ph" "ps")) ("jca" "ph" "ph" "ps" ("id" "ph") "ancli_1")) ;; Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.) (theorem "ancri" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("ancri_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("wa" "ps" "ph")) ("jca" "ph" "ps" "ph" "ancri_1" ("id" "ph"))) ;; Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.) (theorem "ancld" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ancld_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" "ps" ("wa" "ps" "ch"))) ("jcad" "ph" "ps" "ps" "ch" ("idd" "ph" "ps") "ancld_1")) ;; Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.) (theorem "ancrd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ancrd_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" "ps" ("wa" "ch" "ps"))) ("jcad" "ph" "ps" "ch" "ps" "ancrd_1" ("idd" "ph" "ps"))) ;; Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.) (theorem "anc2l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ph" ("wi" "ps" ("wa" "ph" "ch")))) ("biimpi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ph" ("wi" "ps" ("wa" "ph" "ch"))) ("pm5_42" "ph" "ps" "ch"))) ;; Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (theorem "anc2r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ph" ("wi" "ps" ("wa" "ch" "ph")))) ("a2i" "ph" ("wi" "ps" "ch") ("wi" "ps" ("wa" "ch" "ph")) ("imim2d" "ph" "ch" ("wa" "ch" "ph") "ps" ("pm3_21" "ph" "ch")))) ;; Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.) (theorem "anc2li" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anc2li_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" "ps" ("wa" "ph" "ch"))) ("jctild" "ph" "ps" "ch" "ph" "anc2li_1" ("id" "ph"))) ;; Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.) (theorem "anc2ri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anc2ri_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" "ps" ("wa" "ch" "ph"))) ("jctird" "ph" "ps" "ch" "ph" "anc2ri_1" ("id" "ph"))) ;; Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (theorem "pm3_41" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ch") ("wi" ("wa" "ph" "ps") "ch")) ("imim1i" ("wa" "ph" "ps") "ph" "ch" ("simpl" "ph" "ps"))) ;; Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (theorem "pm3_42" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ps" "ch") ("wi" ("wa" "ph" "ps") "ch")) ("imim1i" ("wa" "ph" "ps") "ps" "ch" ("simpr" "ph" "ps"))) ;; Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.) (theorem "pm3_4" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") ("wi" "ph" "ps")) ("a1d" ("wa" "ph" "ps") "ps" "ph" ("simpr" "ph" "ps"))) ;; Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.) (theorem "pm4_45im" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" "ph" ("wa" "ph" ("wi" "ps" "ph"))) ("impbii" "ph" ("wa" "ph" ("wi" "ps" "ph")) ("ancli" "ph" ("wi" "ps" "ph") ("ax_1" "ph" "ps")) ("simpl" "ph" ("wi" "ps" "ph")))) ;; Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.) (theorem "anim12d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("anim12d_1" ("wi" "ph" ("wi" "ps" "ch"))) ("anim12d_2" ("wi" "ph" ("wi" "th" "ta")))) (for) ("wi" "ph" ("wi" ("wa" "ps" "th") ("wa" "ch" "ta"))) ("syl2and" "ph" "ps" "ch" "th" "ta" ("wa" "ch" "ta") "anim12d_1" "anim12d_2" ("idd" "ph" ("wa" "ch" "ta")))) ;; Variant of ~ anim12d where the second implication does not depend on the antecedent. (Contributed by Rodolfo Medina, 12-Oct-2010.) (theorem "anim12d1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("anim12d1_1" ("wi" "ph" ("wi" "ps" "ch"))) ("anim12d1_2" ("wi" "th" "ta"))) (for) ("wi" "ph" ("wi" ("wa" "ps" "th") ("wa" "ch" "ta"))) ("anim12d" "ph" "ps" "ch" "th" "ta" "anim12d1_1" ("a1i" ("wi" "th" "ta") "ph" "anim12d1_2"))) ;; Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.) (theorem "anim1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("anim1d_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" ("wa" "ps" "th") ("wa" "ch" "th"))) ("anim12d" "ph" "ps" "ch" "th" "th" "anim1d_1" ("idd" "ph" "th"))) ;; Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 14-May-1993.) (theorem "anim2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("anim1d_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" ("wa" "th" "ps") ("wa" "th" "ch"))) ("anim12d" "ph" "th" "th" "ps" "ch" ("idd" "ph" "th") "anim1d_1")) ;; Conjoin antecedents and consequents of two premises. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.) (theorem "anim12i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("anim12i_1" ("wi" "ph" "ps")) ("anim12i_2" ("wi" "ch" "th"))) (for) ("wi" ("wa" "ph" "ch") ("wa" "ps" "th")) ("syl2an" "ph" "ps" "th" ("wa" "ps" "th") "ch" "anim12i_1" "anim12i_2" ("id" ("wa" "ps" "th")))) ;; Variant of ~ anim12i with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (theorem "anim12ci" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("anim12i_1" ("wi" "ph" "ps")) ("anim12i_2" ("wi" "ch" "th"))) (for) ("wi" ("wa" "ph" "ch") ("wa" "th" "ps")) ("ancoms" "ch" "ph" ("wa" "th" "ps") ("anim12i" "ch" "th" "ph" "ps" "anim12i_2" "anim12i_1"))) ;; Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.) (theorem "anim1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anim1i_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" "ph" "ch") ("wa" "ps" "ch")) ("anim12i" "ph" "ps" "ch" "ch" "anim1i_1" ("id" "ch"))) ;; Introduce conjunct to both sides of an implication. (Contributed by NM, 3-Jan-1993.) (theorem "anim2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anim1i_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" "ch" "ph") ("wa" "ch" "ps")) ("anim12i" "ch" "ch" "ph" "ps" ("id" "ch") "anim1i_1")) ;; Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.) (theorem "anim12ii" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("anim12ii_1" ("wi" "ph" ("wi" "ps" "ch"))) ("anim12ii_2" ("wi" "th" ("wi" "ps" "ta")))) (for) ("wi" ("wa" "ph" "th") ("wi" "ps" ("wa" "ch" "ta"))) ("jcad" ("wa" "ph" "th") "ps" "ch" "ta" ("adantr" "ph" ("wi" "ps" "ch") "th" "anim12ii_1") ("adantl" "th" ("wi" "ps" "ta") "ph" "anim12ii_2"))) ;; Conjoin antecedents and consequents of two premises. This is the closed theorem form of ~ anim12d . Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it _praeclarum theorema_ (splendid theorem). (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) (theorem "prth" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wi" "ph" "ps") ("wi" "ch" "th")) ("wi" ("wa" "ph" "ch") ("wa" "ps" "th"))) ("anim12d" ("wa" ("wi" "ph" "ps") ("wi" "ch" "th")) "ph" "ps" "ch" "th" ("simpl" ("wi" "ph" "ps") ("wi" "ch" "th")) ("simpr" ("wi" "ph" "ps") ("wi" "ch" "th")))) ;; Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wo" "ph" ("wo" "ps" "ch")) ("wo" "ph" ("wo" "ch" "ps"))) ("orim2i" ("wo" "ps" "ch") ("wo" "ch" "ps") "ph" ("pm1_4" "ps" "ch"))) ;; Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_41" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wo" "ps" ("wo" "ph" "ps")) ("wo" "ph" "ps")) ("jaoi" "ps" ("wo" "ph" "ps") ("wo" "ph" "ps") ("olc" "ps" "ph") ("id" ("wo" "ph" "ps")))) ;; Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_42" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wo" ("wn" "ph") ("wi" "ph" "ps")) ("wi" "ph" "ps")) ("jaoi" ("wn" "ph") ("wi" "ph" "ps") ("wi" "ph" "ps") ("pm2_21" "ph" "ps") ("id" ("wi" "ph" "ps")))) ;; Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_4" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wo" "ph" ("wo" "ph" "ps")) ("wo" "ph" "ps")) ("jaoi" "ph" ("wo" "ph" "ps") ("wo" "ph" "ps") ("orc" "ph" "ps") ("id" ("wo" "ph" "ps")))) ;; Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.) (theorem "pm2_65da" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_65da_1" ("wi" ("wa" "ph" "ps") "ch")) ("pm2_65da_2" ("wi" ("wa" "ph" "ps") ("wn" "ch")))) (for) ("wi" "ph" ("wn" "ps")) ("pm2_65d" "ph" "ps" "ch" ("ex" "ph" "ps" "ch" "pm2_65da_1") ("ex" "ph" "ps" ("wn" "ch") "pm2_65da_2"))) ;; Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_44" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" "ph" ("wo" "ph" ("wa" "ph" "ps"))) ("impbii" "ph" ("wo" "ph" ("wa" "ph" "ps")) ("orc" "ph" ("wa" "ph" "ps")) ("jaoi" "ph" "ph" ("wa" "ph" "ps") ("id" "ph") ("simpl" "ph" "ps")))) ;; Theorem *4.14 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.) (theorem "pm4_14" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" ("wa" "ph" "ps") "ch") ("wi" ("wa" "ph" ("wn" "ch")) ("wn" "ps"))) ("_3bitr4i" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ph" ("wi" ("wn" "ch") ("wn" "ps"))) ("wi" ("wa" "ph" "ps") "ch") ("wi" ("wa" "ph" ("wn" "ch")) ("wn" "ps")) ("imbi2i" ("wi" "ps" "ch") ("wi" ("wn" "ch") ("wn" "ps")) "ph" ("con34b" "ps" "ch")) ("impexp" "ph" "ps" "ch") ("impexp" "ph" ("wn" "ch") ("wn" "ps")))) ;; Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.) (theorem "pm3_37" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" ("wa" "ph" "ps") "ch") ("wi" ("wa" "ph" ("wn" "ch")) ("wn" "ps"))) ("biimpi" ("wi" ("wa" "ph" "ps") "ch") ("wi" ("wa" "ph" ("wn" "ch")) ("wn" "ps")) ("pm4_14" "ph" "ps" "ch"))) ;; Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.) (theorem "nan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wn" ("wa" "ps" "ch"))) ("wi" ("wa" "ph" "ps") ("wn" "ch"))) ("bitr2i" ("wi" ("wa" "ph" "ps") ("wn" "ch")) ("wi" "ph" ("wi" "ps" ("wn" "ch"))) ("wi" "ph" ("wn" ("wa" "ps" "ch"))) ("impexp" "ph" "ps" ("wn" "ch")) ("imbi2i" ("wi" "ps" ("wn" "ch")) ("wn" ("wa" "ps" "ch")) "ph" ("imnan" "ps" "ch")))) ;; Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) (theorem "pm4_15" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" ("wa" "ph" "ps") ("wn" "ch")) ("wi" ("wa" "ps" "ch") ("wn" "ph"))) ("bitr2i" ("wi" ("wa" "ps" "ch") ("wn" "ph")) ("wi" "ph" ("wn" ("wa" "ps" "ch"))) ("wi" ("wa" "ph" "ps") ("wn" "ch")) ("con2b" ("wa" "ps" "ch") "ph") ("nan" "ph" "ps" "ch"))) ;; Implication distributes over disjunction. Theorem *4.78 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2012.) (theorem "pm4_78" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("wi" "ph" ("wo" "ps" "ch"))) ("_3bitr4ri" ("wo" ("wn" "ph") ("wo" "ps" "ch")) ("wo" ("wo" ("wn" "ph") "ps") ("wo" ("wn" "ph") "ch")) ("wi" "ph" ("wo" "ps" "ch")) ("wo" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("orordi" ("wn" "ph") "ps" "ch") ("imor" "ph" ("wo" "ps" "ch")) ("orbi12i" ("wi" "ph" "ps") ("wo" ("wn" "ph") "ps") ("wi" "ph" "ch") ("wo" ("wn" "ph") "ch") ("imor" "ph" "ps") ("imor" "ph" "ch")))) ;; Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.) (theorem "pm4_79" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" ("wi" "ps" "ph") ("wi" "ch" "ph")) ("wi" ("wa" "ps" "ch") "ph")) ("impbii" ("wo" ("wi" "ps" "ph") ("wi" "ch" "ph")) ("wi" ("wa" "ps" "ch") "ph") ("jaoa" ("wi" "ps" "ph") "ps" "ph" ("wi" "ch" "ph") "ch" ("id" ("wi" "ps" "ph")) ("id" ("wi" "ch" "ph"))) ("orrd" ("wi" ("wa" "ps" "ch") "ph") ("wi" "ps" "ph") ("wi" "ch" "ph") ("syl5" ("wn" ("wi" "ps" "ph")) "ps" ("wi" ("wa" "ps" "ch") "ph") ("wi" "ch" "ph") ("simplim" "ps" "ph") ("pm3_3" "ps" "ch" "ph"))))) ;; Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.) (theorem "pm4_87" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wa" ("wa" ("wb" ("wi" ("wa" "ph" "ps") "ch") ("wi" "ph" ("wi" "ps" "ch"))) ("wb" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ps" ("wi" "ph" "ch")))) ("wb" ("wi" "ps" ("wi" "ph" "ch")) ("wi" ("wa" "ps" "ph") "ch"))) ("pm3_2i" ("wa" ("wb" ("wi" ("wa" "ph" "ps") "ch") ("wi" "ph" ("wi" "ps" "ch"))) ("wb" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ps" ("wi" "ph" "ch")))) ("wb" ("wi" "ps" ("wi" "ph" "ch")) ("wi" ("wa" "ps" "ph") "ch")) ("pm3_2i" ("wb" ("wi" ("wa" "ph" "ps") "ch") ("wi" "ph" ("wi" "ps" "ch"))) ("wb" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ps" ("wi" "ph" "ch"))) ("impexp" "ph" "ps" "ch") ("bi2_04" "ph" "ps" "ch")) ("bicomi" ("wi" ("wa" "ps" "ph") "ch") ("wi" "ps" ("wi" "ph" "ch")) ("impexp" "ps" "ph" "ch")))) ;; Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (theorem "pm3_33" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" ("wi" "ph" "ps") ("wi" "ps" "ch")) ("wi" "ph" "ch")) ("imp" ("wi" "ph" "ps") ("wi" "ps" "ch") ("wi" "ph" "ch") ("imim1" "ph" "ps" "ch"))) ;; Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (theorem "pm3_34" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" ("wi" "ps" "ch") ("wi" "ph" "ps")) ("wi" "ph" "ch")) ("imp" ("wi" "ps" "ch") ("wi" "ph" "ps") ("wi" "ph" "ch") ("imim2" "ps" "ch" "ph"))) ;; Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.) (theorem "pm3_35" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("wi" "ph" "ps")) "ps") ("imp" "ph" ("wi" "ph" "ps") "ps" ("pm2_27" "ph" "ps"))) ;; Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_31" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" "ch" ("wi" "ph" "ps")) ("wi" "ph" ("wa" "ps" "ch"))) ("imp" "ch" ("wi" "ph" "ps") ("wi" "ph" ("wa" "ps" "ch")) ("imim2d" "ch" "ps" ("wa" "ps" "ch") "ph" ("pm3_21" "ch" "ps")))) ;; An importation inference. (Contributed by NM, 26-Apr-1994.) Shorten ~ imp4a . (Revised by Wolf Lammen, 19-Jul-2021.) (theorem "imp4b" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imp4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" ("wa" "ph" "ps") ("wi" ("wa" "ch" "th") "ta")) ("impd" ("wa" "ph" "ps") "ch" "th" "ta" ("imp" "ph" "ps" ("wi" "ch" ("wi" "th" "ta")) "imp4_1"))) ;; An importation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Jul-2021.) (theorem "imp4a" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imp4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" "ph" ("wi" "ps" ("wi" ("wa" "ch" "th") "ta"))) ("ex" "ph" "ps" ("wi" ("wa" "ch" "th") "ta") ("imp4b" "ph" "ps" "ch" "th" "ta" "imp4_1"))) ;; Obsolete proof of ~ imp4a as of 19-Jul-2021. (Contributed by NM, 26-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "imp4aOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imp4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" "ph" ("wi" "ps" ("wi" ("wa" "ch" "th") "ta"))) ("syl6ibr" "ph" "ps" ("wi" "ch" ("wi" "th" "ta")) ("wi" ("wa" "ch" "th") "ta") "imp4_1" ("impexp" "ch" "th" "ta"))) ;; Obsolete proof of ~ imp4b as of 19-Jul-2021. (Contributed by NM, 26-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "imp4bOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imp4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" ("wa" "ph" "ps") ("wi" ("wa" "ch" "th") "ta")) ("imp" "ph" "ps" ("wi" ("wa" "ch" "th") "ta") ("imp4a" "ph" "ps" "ch" "th" "ta" "imp4_1"))) ;; An importation inference. (Contributed by NM, 26-Apr-1994.) (theorem "imp4c" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imp4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" "ph" ("wi" ("wa" ("wa" "ps" "ch") "th") "ta")) ("impd" "ph" ("wa" "ps" "ch") "th" "ta" ("impd" "ph" "ps" "ch" ("wi" "th" "ta") "imp4_1"))) ;; An importation inference. (Contributed by NM, 26-Apr-1994.) (theorem "imp4d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imp4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" "ph" ("wi" ("wa" "ps" ("wa" "ch" "th")) "ta")) ("impd" "ph" "ps" ("wa" "ch" "th") "ta" ("imp4a" "ph" "ps" "ch" "th" "ta" "imp4_1"))) ;; An importation inference. (Contributed by NM, 26-Apr-1994.) (theorem "imp41" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imp4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") ("imp31" ("wa" "ph" "ps") "ch" "th" "ta" ("imp" "ph" "ps" ("wi" "ch" ("wi" "th" "ta")) "imp4_1"))) ;; An importation inference. (Contributed by NM, 26-Apr-1994.) (theorem "imp42" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imp4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" ("wa" ("wa" "ph" ("wa" "ps" "ch")) "th") "ta") ("imp" ("wa" "ph" ("wa" "ps" "ch")) "th" "ta" ("imp32" "ph" "ps" "ch" ("wi" "th" "ta") "imp4_1"))) ;; An importation inference. (Contributed by NM, 26-Apr-1994.) (theorem "imp43" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imp4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) "ta") ("imp" ("wa" "ph" "ps") ("wa" "ch" "th") "ta" ("imp4b" "ph" "ps" "ch" "th" "ta" "imp4_1"))) ;; An importation inference. (Contributed by NM, 26-Apr-1994.) (theorem "imp44" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imp4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" ("wa" "ph" ("wa" ("wa" "ps" "ch") "th")) "ta") ("imp" "ph" ("wa" ("wa" "ps" "ch") "th") "ta" ("imp4c" "ph" "ps" "ch" "th" "ta" "imp4_1"))) ;; An importation inference. (Contributed by NM, 26-Apr-1994.) (theorem "imp45" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("imp4_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" ("wa" "ph" ("wa" "ps" ("wa" "ch" "th"))) "ta") ("imp" "ph" ("wa" "ps" ("wa" "ch" "th")) "ta" ("imp4d" "ph" "ps" "ch" "th" "ta" "imp4_1"))) ;; An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) (theorem "imp5a" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("imp5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" ("wa" "th" "ta") "et")))) ("syl8" "ph" "ps" "ch" ("wi" "th" ("wi" "ta" "et")) ("wi" ("wa" "th" "ta") "et") "imp5_1" ("pm3_31" "th" "ta" "et"))) ;; An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) (theorem "imp5d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("imp5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" ("wa" ("wa" "ph" "ps") "ch") ("wi" ("wa" "th" "ta") "et")) ("impd" ("wa" ("wa" "ph" "ps") "ch") "th" "ta" "et" ("imp31" "ph" "ps" "ch" ("wi" "th" ("wi" "ta" "et")) "imp5_1"))) ;; An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) (theorem "imp5g" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("imp5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" ("wa" "ph" "ps") ("wi" ("wa" ("wa" "ch" "th") "ta") "et")) ("imp4c" ("wa" "ph" "ps") "ch" "th" "ta" "et" ("imp" "ph" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))) "imp5_1"))) ;; An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) (theorem "imp55" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("imp5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" ("wa" ("wa" "ph" ("wa" "ps" ("wa" "ch" "th"))) "ta") "et") ("imp42" "ph" "ps" ("wa" "ch" "th") "ta" "et" ("imp4a" "ph" "ps" "ch" "th" ("wi" "ta" "et") "imp5_1"))) ;; An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) (theorem "imp511" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("imp5_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))))) (for) ("wi" ("wa" "ph" ("wa" ("wa" "ps" ("wa" "ch" "th")) "ta")) "et") ("imp44" "ph" "ps" ("wa" "ch" "th") "ta" "et" ("imp4a" "ph" "ps" "ch" "th" ("wi" "ta" "et") "imp5_1"))) ;; Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.) (theorem "expimpd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("expimpd_1" ("wi" ("wa" "ph" "ps") ("wi" "ch" "th")))) (for) ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")) ("impd" "ph" "ps" "ch" "th" ("ex" "ph" "ps" ("wi" "ch" "th") "expimpd_1"))) ;; An exportation inference. (Contributed by NM, 26-Apr-1994.) (theorem "exp31" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("exp31_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))) ("ex" "ph" "ps" ("wi" "ch" "th") ("ex" ("wa" "ph" "ps") "ch" "th" "exp31_1"))) ;; An exportation inference. (Contributed by NM, 26-Apr-1994.) (theorem "exp32" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("exp32_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))) ("expd" "ph" "ps" "ch" "th" ("ex" "ph" ("wa" "ps" "ch") "th" "exp32_1"))) ;; An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) Shorten ~ exp4a . (Revised by Wolf Lammen, 20-Jul-2021.) (theorem "exp4b" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("exp4b_1" ("wi" ("wa" "ph" "ps") ("wi" ("wa" "ch" "th") "ta")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("ex" "ph" "ps" ("wi" "ch" ("wi" "th" "ta")) ("expd" ("wa" "ph" "ps") "ch" "th" "ta" "exp4b_1"))) ;; An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jul-2021.) (theorem "exp4a" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("exp4a_1" ("wi" "ph" ("wi" "ps" ("wi" ("wa" "ch" "th") "ta"))))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("exp4b" "ph" "ps" "ch" "th" "ta" ("imp" "ph" "ps" ("wi" ("wa" "ch" "th") "ta") "exp4a_1"))) ;; Obsolete proof of ~ exp4a as of 20-Jul-2021. (Contributed by NM, 26-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "exp4aOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("exp4aOLD_1" ("wi" "ph" ("wi" "ps" ("wi" ("wa" "ch" "th") "ta"))))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("syl6ib" "ph" "ps" ("wi" ("wa" "ch" "th") "ta") ("wi" "ch" ("wi" "th" "ta")) "exp4aOLD_1" ("impexp" "ch" "th" "ta"))) ;; Obsolete proof of ~ exp4b as of 20-Jul-2021. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "exp4bOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("exp4bOLD_1" ("wi" ("wa" "ph" "ps") ("wi" ("wa" "ch" "th") "ta")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("exp4a" "ph" "ps" "ch" "th" "ta" ("ex" "ph" "ps" ("wi" ("wa" "ch" "th") "ta") "exp4bOLD_1"))) ;; An exportation inference. (Contributed by NM, 26-Apr-1994.) (theorem "exp4c" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("exp4c_1" ("wi" "ph" ("wi" ("wa" ("wa" "ps" "ch") "th") "ta")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("expd" "ph" "ps" "ch" ("wi" "th" "ta") ("expd" "ph" ("wa" "ps" "ch") "th" "ta" "exp4c_1"))) ;; An exportation inference. (Contributed by NM, 26-Apr-1994.) (theorem "exp4d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("exp4d_1" ("wi" "ph" ("wi" ("wa" "ps" ("wa" "ch" "th")) "ta")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("exp4a" "ph" "ps" "ch" "th" "ta" ("expd" "ph" "ps" ("wa" "ch" "th") "ta" "exp4d_1"))) ;; An exportation inference. (Contributed by NM, 26-Apr-1994.) (theorem "exp41" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("exp41_1" ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("exp31" "ph" "ps" "ch" ("wi" "th" "ta") ("ex" ("wa" ("wa" "ph" "ps") "ch") "th" "ta" "exp41_1"))) ;; An exportation inference. (Contributed by NM, 26-Apr-1994.) (theorem "exp42" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("exp42_1" ("wi" ("wa" ("wa" "ph" ("wa" "ps" "ch")) "th") "ta"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("expd" "ph" "ps" "ch" ("wi" "th" "ta") ("exp31" "ph" ("wa" "ps" "ch") "th" "ta" "exp42_1"))) ;; An exportation inference. (Contributed by NM, 26-Apr-1994.) (theorem "exp43" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("exp43_1" ("wi" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) "ta"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("exp4b" "ph" "ps" "ch" "th" "ta" ("ex" ("wa" "ph" "ps") ("wa" "ch" "th") "ta" "exp43_1"))) ;; An exportation inference. (Contributed by NM, 26-Apr-1994.) (theorem "exp44" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("exp44_1" ("wi" ("wa" "ph" ("wa" ("wa" "ps" "ch") "th")) "ta"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("expd" "ph" "ps" "ch" ("wi" "th" "ta") ("exp32" "ph" ("wa" "ps" "ch") "th" "ta" "exp44_1"))) ;; An exportation inference. (Contributed by NM, 26-Apr-1994.) (theorem "exp45" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("exp45_1" ("wi" ("wa" "ph" ("wa" "ps" ("wa" "ch" "th"))) "ta"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("exp4a" "ph" "ps" "ch" "th" "ta" ("exp32" "ph" "ps" ("wa" "ch" "th") "ta" "exp45_1"))) ;; Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.) (theorem "expr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("expr_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ph" "ps") ("wi" "ch" "th")) ("imp" "ph" "ps" ("wi" "ch" "th") ("exp32" "ph" "ps" "ch" "th" "expr_1"))) ;; An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) (theorem "exp5c" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("exp5c_1" ("wi" "ph" ("wi" ("wa" "ps" "ch") ("wi" ("wa" "th" "ta") "et"))))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))) ("expd" "ph" "ps" "ch" ("wi" "th" ("wi" "ta" "et")) ("exp4a" "ph" ("wa" "ps" "ch") "th" "ta" "et" "exp5c_1"))) ;; An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) (theorem "exp5j" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("exp5j_1" ("wi" "ph" ("wi" ("wa" ("wa" ("wa" "ps" "ch") "th") "ta") "et")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))) ("exp4c" "ph" "ps" "ch" "th" ("wi" "ta" "et") ("expd" "ph" ("wa" ("wa" "ps" "ch") "th") "ta" "et" "exp5j_1"))) ;; An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) (theorem "exp5l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("exp5l_1" ("wi" "ph" ("wi" ("wa" ("wa" "ps" "ch") ("wa" "th" "ta")) "et")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))) ("exp5c" "ph" "ps" "ch" "th" "ta" "et" ("expd" "ph" ("wa" "ps" "ch") ("wa" "th" "ta") "et" "exp5l_1"))) ;; An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.) (theorem "exp53" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("exp53_1" ("wi" ("wa" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) "ta") "et"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))) ("exp43" "ph" "ps" "ch" "th" ("wi" "ta" "et") ("ex" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) "ta" "et" "exp53_1"))) ;; Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.) (theorem "expl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("expl_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")) ("impd" "ph" "ps" "ch" "th" ("exp31" "ph" "ps" "ch" "th" "expl_1"))) ;; Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.) (theorem "impr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("impr_1" ("wi" ("wa" "ph" "ps") ("wi" "ch" "th")))) (for) ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th") ("imp32" "ph" "ps" "ch" "th" ("ex" "ph" "ps" ("wi" "ch" "th") "impr_1"))) ;; Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.) (theorem "impl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("impl_1" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" ("wa" ("wa" "ph" "ps") "ch") "th") ("imp31" "ph" "ps" "ch" "th" ("expd" "ph" "ps" "ch" "th" "impl_1"))) ;; Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.) (theorem "impac" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("impac_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" ("wa" "ph" "ps") ("wa" "ch" "ps")) ("imp" "ph" "ps" ("wa" "ch" "ps") ("ancrd" "ph" "ps" "ch" "impac_1"))) ;; Inference form of ~ exbir . This proof is ~ exbiriVD automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.) (theorem "exbiri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("exbiri_1" ("wi" ("wa" "ph" "ps") ("wb" "ch" "th")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "th" "ch"))) ("exp31" "ph" "ps" "th" "ch" ("biimpar" ("wa" "ph" "ps") "ch" "th" "exbiri_1"))) ;; Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) (theorem "simprbda" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm3_26bda_1" ("wi" "ph" ("wb" "ps" ("wa" "ch" "th"))))) (for) ("wi" ("wa" "ph" "ps") "ch") ("simpld" ("wa" "ph" "ps") "ch" "th" ("biimpa" "ph" "ps" ("wa" "ch" "th") "pm3_26bda_1"))) ;; Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) (theorem "simplbda" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm3_26bda_1" ("wi" "ph" ("wb" "ps" ("wa" "ch" "th"))))) (for) ("wi" ("wa" "ph" "ps") "th") ("simprd" ("wa" "ph" "ps") "ch" "th" ("biimpa" "ph" "ps" ("wa" "ch" "th") "pm3_26bda_1"))) ;; Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.) (theorem "simplbi2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("simplbi2_1" ("wb" "ph" ("wa" "ps" "ch")))) (for) ("wi" "ps" ("wi" "ch" "ph")) ("ex" "ps" "ch" "ph" ("biimpri" "ph" ("wa" "ps" "ch") "simplbi2_1"))) ;; Closed form of ~ simplbi2com . (Contributed by Alan Sare, 22-Jul-2012.) (theorem "simplbi2comt" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wb" "ph" ("wa" "ps" "ch")) ("wi" "ch" ("wi" "ps" "ph"))) ("expcomd" ("wb" "ph" ("wa" "ps" "ch")) "ps" "ch" "ph" ("biimpr" "ph" ("wa" "ps" "ch")))) ;; A deduction eliminating a conjunct, similar to ~ simplbi2 . (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.) (theorem "simplbi2com" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("simplbi2com_1" ("wb" "ph" ("wa" "ps" "ch")))) (for) ("wi" "ch" ("wi" "ps" "ph")) ("com12" "ps" "ch" "ph" ("simplbi2" "ph" "ps" "ch" "simplbi2com_1"))) ;; Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.) (Proof shortened by Wolf Lammen, 26-Mar-2022.) (theorem "simpl2im" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("simpl2im_1" ("wi" "ph" ("wa" "ps" "ch"))) ("simpl2im_2" ("wi" "ch" "th"))) (for) ("wi" "ph" "th") ("syl" "ph" "ch" "th" ("simprd" "ph" "ps" "ch" "simpl2im_1") "simpl2im_2")) ;; Obsolete proof of ~ simpl2im as of 26-Mar-2022. (Contributed by BJ/AV, 5-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.) (theorem "simpl2imOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("simpl2im_1" ("wi" "ph" ("wa" "ps" "ch"))) ("simpl2im_2" ("wi" "ch" "th"))) (for) ("wi" "ph" "th") ("_3syl" "ph" ("wa" "ps" "ch") "ch" "th" "simpl2im_1" ("simpr" "ps" "ch") "simpl2im_2")) ;; Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 26-Mar-2022.) (theorem "simplbiim" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("simplbiim_1" ("wb" "ph" ("wa" "ps" "ch"))) ("simplbiim_2" ("wi" "ch" "th"))) (for) ("wi" "ph" "th") ("syl" "ph" "ch" "th" ("simprbi" "ph" "ps" "ch" "simplbiim_1") "simplbiim_2")) ;; Obsolete proof of ~ simplbiim as of 26-Mar-2022. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.) (theorem "simplbiimOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("simplbiim_1" ("wb" "ph" ("wa" "ps" "ch"))) ("simplbiim_2" ("wi" "ch" "th"))) (for) ("wi" "ph" "th") ("sylbi" "ph" ("wa" "ps" "ch") "th" "simplbiim_1" ("adantl" "ch" "th" "ps" "simplbiim_2"))) ;; A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 24-Jan-1993.) (theorem "dfbi2" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" "ph" "ps") ("wa" ("wi" "ph" "ps") ("wi" "ps" "ph"))) ("bitr4i" ("wb" "ph" "ps") ("wn" ("wi" ("wi" "ph" "ps") ("wn" ("wi" "ps" "ph")))) ("wa" ("wi" "ph" "ps") ("wi" "ps" "ph")) ("dfbi1" "ph" "ps") ("df_an" ("wi" "ph" "ps") ("wi" "ps" "ph")))) ;; Definition ~ df-bi rewritten in an abbreviated form to help intuitive understanding of that definition. Note that it is a conjunction of two implications; one which asserts properties that follow from the biconditional and one which asserts properties that imply the biconditional. (Contributed by NM, 15-Aug-2008.) (theorem "dfbi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wa" ("wi" ("wb" "ph" "ps") ("wa" ("wi" "ph" "ps") ("wi" "ps" "ph"))) ("wi" ("wa" ("wi" "ph" "ps") ("wi" "ps" "ph")) ("wb" "ph" "ps"))) ("pm3_2i" ("wi" ("wb" "ph" "ps") ("wa" ("wi" "ph" "ps") ("wi" "ps" "ph"))) ("wi" ("wa" ("wi" "ph" "ps") ("wi" "ps" "ph")) ("wb" "ph" "ps")) ("biimpi" ("wb" "ph" "ps") ("wa" ("wi" "ph" "ps") ("wi" "ps" "ph")) ("dfbi2" "ph" "ps")) ("biimpri" ("wb" "ph" "ps") ("wa" ("wi" "ph" "ps") ("wi" "ps" "ph")) ("dfbi2" "ph" "ps")))) ;; Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.) (theorem "pm4_71" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" "ps") ("wb" "ph" ("wa" "ph" "ps"))) ("_3bitr4i" ("wi" "ph" ("wa" "ph" "ps")) ("wa" ("wi" "ph" ("wa" "ph" "ps")) ("wi" ("wa" "ph" "ps") "ph")) ("wi" "ph" "ps") ("wb" "ph" ("wa" "ph" "ps")) ("biantru" ("wi" ("wa" "ph" "ps") "ph") ("wi" "ph" ("wa" "ph" "ps")) ("simpl" "ph" "ps")) ("anclb" "ph" "ps") ("dfbi2" "ph" ("wa" "ph" "ps")))) ;; Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.) (theorem "pm4_71r" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" "ps") ("wb" "ph" ("wa" "ps" "ph"))) ("bitri" ("wi" "ph" "ps") ("wb" "ph" ("wa" "ph" "ps")) ("wb" "ph" ("wa" "ps" "ph")) ("pm4_71" "ph" "ps") ("bibi2i" ("wa" "ph" "ps") ("wa" "ps" "ph") "ph" ("ancom" "ph" "ps")))) ;; Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 4-Jan-2004.) (theorem "pm4_71i" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm4_71i_1" ("wi" "ph" "ps"))) (for) ("wb" "ph" ("wa" "ph" "ps")) ("mpbi" ("wi" "ph" "ps") ("wb" "ph" ("wa" "ph" "ps")) "pm4_71i_1" ("pm4_71" "ph" "ps"))) ;; Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 1-Dec-2003.) (theorem "pm4_71ri" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm4_71ri_1" ("wi" "ph" "ps"))) (for) ("wb" "ph" ("wa" "ps" "ph")) ("mpbi" ("wi" "ph" "ps") ("wb" "ph" ("wa" "ps" "ph")) "pm4_71ri_1" ("pm4_71r" "ph" "ps"))) ;; Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by Mario Carneiro, 25-Dec-2016.) (theorem "pm4_71d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm4_71rd_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wb" "ps" ("wa" "ps" "ch"))) ("sylib" "ph" ("wi" "ps" "ch") ("wb" "ps" ("wa" "ps" "ch")) "pm4_71rd_1" ("pm4_71" "ps" "ch"))) ;; Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 10-Feb-2005.) (theorem "pm4_71rd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm4_71rd_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wb" "ps" ("wa" "ch" "ps"))) ("sylib" "ph" ("wi" "ps" "ch") ("wb" "ps" ("wa" "ch" "ps")) "pm4_71rd_1" ("pm4_71r" "ps" "ch"))) ;; Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (theorem "pm5_32" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wb" "ps" "ch")) ("wb" ("wa" "ph" "ps") ("wa" "ph" "ch"))) ("bitr4i" ("wi" "ph" ("wb" "ps" "ch")) ("wb" ("wn" ("wi" "ph" ("wn" "ps"))) ("wn" ("wi" "ph" ("wn" "ch")))) ("wb" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("_3bitri" ("wi" "ph" ("wb" "ps" "ch")) ("wi" "ph" ("wb" ("wn" "ps") ("wn" "ch"))) ("wb" ("wi" "ph" ("wn" "ps")) ("wi" "ph" ("wn" "ch"))) ("wb" ("wn" ("wi" "ph" ("wn" "ps"))) ("wn" ("wi" "ph" ("wn" "ch")))) ("imbi2i" ("wb" "ps" "ch") ("wb" ("wn" "ps") ("wn" "ch")) "ph" ("notbi" "ps" "ch")) ("pm5_74" "ph" ("wn" "ps") ("wn" "ch")) ("notbi" ("wi" "ph" ("wn" "ps")) ("wi" "ph" ("wn" "ch")))) ("bibi12i" ("wa" "ph" "ps") ("wn" ("wi" "ph" ("wn" "ps"))) ("wa" "ph" "ch") ("wn" ("wi" "ph" ("wn" "ch"))) ("df_an" "ph" "ps") ("df_an" "ph" "ch")))) ;; Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.) (theorem "pm5_32i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm5_32i_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wb" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("mpbi" ("wi" "ph" ("wb" "ps" "ch")) ("wb" ("wa" "ph" "ps") ("wa" "ph" "ch")) "pm5_32i_1" ("pm5_32" "ph" "ps" "ch"))) ;; Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.) (theorem "pm5_32ri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm5_32i_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wb" ("wa" "ps" "ph") ("wa" "ch" "ph")) ("_3bitr4i" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ps" "ph") ("wa" "ch" "ph") ("pm5_32i" "ph" "ps" "ch" "pm5_32i_1") ("ancom" "ps" "ph") ("ancom" "ch" "ph"))) ;; Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.) (theorem "pm5_32d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm5_32d_1" ("wi" "ph" ("wi" "ps" ("wb" "ch" "th"))))) (for) ("wi" "ph" ("wb" ("wa" "ps" "ch") ("wa" "ps" "th"))) ("sylib" "ph" ("wi" "ps" ("wb" "ch" "th")) ("wb" ("wa" "ps" "ch") ("wa" "ps" "th")) "pm5_32d_1" ("pm5_32" "ps" "ch" "th"))) ;; Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.) (theorem "pm5_32rd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm5_32d_1" ("wi" "ph" ("wi" "ps" ("wb" "ch" "th"))))) (for) ("wi" "ph" ("wb" ("wa" "ch" "ps") ("wa" "th" "ps"))) ("_3bitr4g" "ph" ("wa" "ps" "ch") ("wa" "ps" "th") ("wa" "ch" "ps") ("wa" "th" "ps") ("pm5_32d" "ph" "ps" "ch" "th" "pm5_32d_1") ("ancom" "ch" "ps") ("ancom" "th" "ps"))) ;; Distribution of implication over biconditional (deduction rule). (Contributed by NM, 9-Dec-2006.) (theorem "pm5_32da" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm5_32da_1" ("wi" ("wa" "ph" "ps") ("wb" "ch" "th")))) (for) ("wi" "ph" ("wb" ("wa" "ps" "ch") ("wa" "ps" "th"))) ("pm5_32d" "ph" "ps" "ch" "th" ("ex" "ph" "ps" ("wb" "ch" "th") "pm5_32da_1"))) ;; Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (theorem "biadan2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("biadan2_1" ("wi" "ph" "ps")) ("biadan2_2" ("wi" "ps" ("wb" "ph" "ch")))) (for) ("wb" "ph" ("wa" "ps" "ch")) ("bitri" "ph" ("wa" "ps" "ph") ("wa" "ps" "ch") ("pm4_71ri" "ph" "ps" "biadan2_1") ("pm5_32i" "ps" "ph" "ch" "biadan2_2"))) ;; Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.) (theorem "pm4_24" (for ("ph" ( "wff"))) (for) (for) ("wb" "ph" ("wa" "ph" "ph")) ("pm4_71i" "ph" "ph" ("id" "ph"))) ;; Idempotent law for conjunction. (Contributed by NM, 8-Jan-2004.) (Proof shortened by Wolf Lammen, 14-Mar-2014.) (theorem "anidm" (for ("ph" ( "wff"))) (for) (for) ("wb" ("wa" "ph" "ph") "ph") ("bicomi" "ph" ("wa" "ph" "ph") ("pm4_24" "ph"))) ;; Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.) (theorem "anidms" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("anidms_1" ("wi" ("wa" "ph" "ph") "ps"))) (for) ("wi" "ph" "ps") ("pm2_43i" "ph" "ps" ("ex" "ph" "ph" "ps" "anidms_1"))) ;; Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.) (theorem "anidmdbi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wa" "ps" "ps")) ("wi" "ph" "ps")) ("imbi2i" ("wa" "ps" "ps") "ps" "ph" ("anidm" "ps"))) ;; Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.) (theorem "anasss" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("anasss_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th") ("imp32" "ph" "ps" "ch" "th" ("exp31" "ph" "ps" "ch" "th" "anasss_1"))) ;; Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.) (theorem "anassrs" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("anassrs_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" ("wa" "ph" "ps") "ch") "th") ("imp31" "ph" "ps" "ch" "th" ("exp32" "ph" "ps" "ch" "th" "anassrs_1"))) ;; Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) (theorem "anass" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" ("wa" "ph" "ps") "ch") ("wa" "ph" ("wa" "ps" "ch"))) ("impbii" ("wa" ("wa" "ph" "ps") "ch") ("wa" "ph" ("wa" "ps" "ch")) ("anassrs" "ph" "ps" "ch" ("wa" "ph" ("wa" "ps" "ch")) ("id" ("wa" "ph" ("wa" "ps" "ch")))) ("anasss" "ph" "ps" "ch" ("wa" ("wa" "ph" "ps") "ch") ("id" ("wa" ("wa" "ph" "ps") "ch"))))) ;; A syllogism inference. (Contributed by NM, 10-Mar-2005.) (theorem "sylanl1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylanl1_1" ("wi" "ph" "ps")) ("sylanl1_2" ("wi" ("wa" ("wa" "ps" "ch") "th") "ta"))) (for) ("wi" ("wa" ("wa" "ph" "ch") "th") "ta") ("sylan" ("wa" "ph" "ch") ("wa" "ps" "ch") "th" "ta" ("anim1i" "ph" "ps" "ch" "sylanl1_1") "sylanl1_2")) ;; A syllogism inference. (Contributed by NM, 1-Jan-2005.) (theorem "sylanl2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylanl2_1" ("wi" "ph" "ch")) ("sylanl2_2" ("wi" ("wa" ("wa" "ps" "ch") "th") "ta"))) (for) ("wi" ("wa" ("wa" "ps" "ph") "th") "ta") ("sylan" ("wa" "ps" "ph") ("wa" "ps" "ch") "th" "ta" ("anim2i" "ph" "ch" "ps" "sylanl2_1") "sylanl2_2")) ;; A syllogism inference. (Contributed by NM, 9-Apr-2005.) (theorem "sylanr1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylanr1_1" ("wi" "ph" "ch")) ("sylanr1_2" ("wi" ("wa" "ps" ("wa" "ch" "th")) "ta"))) (for) ("wi" ("wa" "ps" ("wa" "ph" "th")) "ta") ("sylan2" ("wa" "ph" "th") "ps" ("wa" "ch" "th") "ta" ("anim1i" "ph" "ch" "th" "sylanr1_1") "sylanr1_2")) ;; A syllogism inference. (Contributed by NM, 9-Apr-2005.) (theorem "sylanr2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylanr2_1" ("wi" "ph" "th")) ("sylanr2_2" ("wi" ("wa" "ps" ("wa" "ch" "th")) "ta"))) (for) ("wi" ("wa" "ps" ("wa" "ch" "ph")) "ta") ("sylan2" ("wa" "ch" "ph") "ps" ("wa" "ch" "th") "ta" ("anim2i" "ph" "th" "ch" "sylanr2_1") "sylanr2_2")) ;; A syllogism inference. (Contributed by NM, 2-May-1996.) (theorem "sylani" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylani_1" ("wi" "ph" "ch")) ("sylani_2" ("wi" "ps" ("wi" ("wa" "ch" "th") "ta")))) (for) ("wi" "ps" ("wi" ("wa" "ph" "th") "ta")) ("syland" "ps" "ph" "ch" "th" "ta" ("a1i" ("wi" "ph" "ch") "ps" "sylani_1") "sylani_2")) ;; A syllogism inference. (Contributed by NM, 1-Aug-1994.) (theorem "sylan2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylan2i_1" ("wi" "ph" "th")) ("sylan2i_2" ("wi" "ps" ("wi" ("wa" "ch" "th") "ta")))) (for) ("wi" "ps" ("wi" ("wa" "ch" "ph") "ta")) ("sylan2d" "ps" "ph" "th" "ch" "ta" ("a1i" ("wi" "ph" "th") "ps" "sylan2i_1") "sylan2i_2")) ;; A syllogism inference. (Contributed by NM, 3-Aug-1999.) (theorem "syl2ani" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl2ani_1" ("wi" "ph" "ch")) ("syl2ani_2" ("wi" "et" "th")) ("syl2ani_3" ("wi" "ps" ("wi" ("wa" "ch" "th") "ta")))) (for) ("wi" "ps" ("wi" ("wa" "ph" "et") "ta")) ("sylani" "ph" "ps" "ch" "et" "ta" "syl2ani_1" ("sylan2i" "et" "ps" "ch" "th" "ta" "syl2ani_2" "syl2ani_3"))) ;; Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "sylan9" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylan9_1" ("wi" "ph" ("wi" "ps" "ch"))) ("sylan9_2" ("wi" "th" ("wi" "ch" "ta")))) (for) ("wi" ("wa" "ph" "th") ("wi" "ps" "ta")) ("imp" "ph" "th" ("wi" "ps" "ta") ("syl9" "ph" "ps" "ch" "th" "ta" "sylan9_1" "sylan9_2"))) ;; Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.) (theorem "sylan9r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylan9r_1" ("wi" "ph" ("wi" "ps" "ch"))) ("sylan9r_2" ("wi" "th" ("wi" "ch" "ta")))) (for) ("wi" ("wa" "th" "ph") ("wi" "ps" "ta")) ("imp" "th" "ph" ("wi" "ps" "ta") ("syl9r" "ph" "ps" "ch" "th" "ta" "sylan9r_1" "sylan9r_2"))) ;; A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.) (theorem "mtand" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mtand_1" ("wi" "ph" ("wn" "ch"))) ("mtand_2" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" "ph" ("wn" "ps")) ("mtod" "ph" "ps" "ch" "mtand_1" ("ex" "ph" "ps" "ch" "mtand_2"))) ;; A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) (theorem "mtord" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mtord_1" ("wi" "ph" ("wn" "ch"))) ("mtord_2" ("wi" "ph" ("wn" "th"))) ("mtord_3" ("wi" "ph" ("wi" "ps" ("wo" "ch" "th"))))) (for) ("wi" "ph" ("wn" "ps")) ("mtod" "ph" "ps" "th" "mtord_2" ("mpid" "ph" "ps" ("wn" "ch") "th" "mtord_1" ("syl6ib" "ph" "ps" ("wo" "ch" "th") ("wi" ("wn" "ch") "th") "mtord_3" ("df_or" "ch" "th"))))) ;; Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.) (theorem "syl2anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("syl2anc_1" ("wi" "ph" "ps")) ("syl2anc_2" ("wi" "ph" "ch")) ("syl2anc_3" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" "ph" "th") ("sylc" "ph" "ps" "ch" "th" "syl2anc_1" "syl2anc_2" ("ex" "ps" "ch" "th" "syl2anc_3"))) ;; Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.) (theorem "sylancl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylancl_1" ("wi" "ph" "ps")) ("sylancl_2" "ch") ("sylancl_3" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" "ph" "th") ("syl2anc" "ph" "ps" "ch" "th" "sylancl_1" ("a1i" "ch" "ph" "sylancl_2") "sylancl_3")) ;; Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.) (theorem "sylancr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylancr_1" "ps") ("sylancr_2" ("wi" "ph" "ch")) ("sylancr_3" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" "ph" "th") ("syl2anc" "ph" "ps" "ch" "th" ("a1i" "ps" "ph" "sylancr_1") "sylancr_2" "sylancr_3")) ;; Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.) (theorem "sylanblc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylanblc_1" ("wi" "ph" "ps")) ("sylanblc_2" "ch") ("sylanblc_3" ("wb" ("wa" "ps" "ch") "th"))) (for) ("wi" "ph" "th") ("sylancl" "ph" "ps" "ch" "th" "sylanblc_1" "sylanblc_2" ("biimpi" ("wa" "ps" "ch") "th" "sylanblc_3"))) ;; Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.) (theorem "sylanblrc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylanblrc_1" ("wi" "ph" "ps")) ("sylanblrc_2" "ch") ("sylanblrc_3" ("wb" "th" ("wa" "ps" "ch")))) (for) ("wi" "ph" "th") ("sylancl" "ph" "ps" "ch" "th" "sylanblrc_1" "sylanblrc_2" ("biimpri" "th" ("wa" "ps" "ch") "sylanblrc_3"))) ;; Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.) (theorem "sylanbrc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylanbrc_1" ("wi" "ph" "ps")) ("sylanbrc_2" ("wi" "ph" "ch")) ("sylanbrc_3" ("wb" "th" ("wa" "ps" "ch")))) (for) ("wi" "ph" "th") ("sylibr" "ph" ("wa" "ps" "ch") "th" ("jca" "ph" "ps" "ch" "sylanbrc_1" "sylanbrc_2") "sylanbrc_3")) ;; A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.) (theorem "sylancb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylancb_1" ("wb" "ph" "ps")) ("sylancb_2" ("wb" "ph" "ch")) ("sylancb_3" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" "ph" "th") ("anidms" "ph" "th" ("syl2anb" "ph" "ps" "ch" "th" "ph" "sylancb_1" "sylancb_2" "sylancb_3"))) ;; A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.) (theorem "sylancbr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylancbr_1" ("wb" "ps" "ph")) ("sylancbr_2" ("wb" "ch" "ph")) ("sylancbr_3" ("wi" ("wa" "ps" "ch") "th"))) (for) ("wi" "ph" "th") ("anidms" "ph" "th" ("syl2anbr" "ph" "ps" "ch" "th" "ph" "sylancbr_1" "sylancbr_2" "sylancbr_3"))) ;; Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008.) (theorem "sylancom" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("sylancom_1" ("wi" ("wa" "ph" "ps") "ch")) ("sylancom_2" ("wi" ("wa" "ch" "ps") "th"))) (for) ("wi" ("wa" "ph" "ps") "th") ("syl2anc" ("wa" "ph" "ps") "ch" "ps" "th" "sylancom_1" ("simpr" "ph" "ps") "sylancom_2")) ;; An inference based on modus ponens. (Contributed by NM, 23-May-1999.) (Proof shortened by Wolf Lammen, 22-Nov-2012.) (theorem "mpdan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpdan_1" ("wi" "ph" "ps")) ("mpdan_2" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" "ph" "ch") ("syl2anc" "ph" "ph" "ps" "ch" ("id" "ph") "mpdan_1" "mpdan_2")) ;; An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) (theorem "mpancom" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpancom_1" ("wi" "ps" "ph")) ("mpancom_2" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" "ps" "ch") ("syl2anc" "ps" "ph" "ps" "ch" "mpancom_1" ("id" "ps") "mpancom_2")) ;; A deduction which "stacks" a hypothesis. (Contributed by Stanislas Polu, 9-Mar-2020.) (Proof shortened by Wolf Lammen, 28-Mar-2021.) (theorem "mpidan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpidan_1" ("wi" "ph" "ch")) ("mpidan_2" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" "ph" "ps") "th") ("mpdan" ("wa" "ph" "ps") "ch" "th" ("adantr" "ph" "ch" "ps" "mpidan_1") "mpidan_2")) ;; An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) (theorem "mpan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpan_1" "ph") ("mpan_2" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" "ps" "ch") ("mpancom" "ph" "ps" "ch" ("a1i" "ph" "ps" "mpan_1") "mpan_2")) ;; An inference based on modus ponens. (Contributed by NM, 16-Sep-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2012.) (theorem "mpan2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpan2_1" "ps") ("mpan2_2" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" "ph" "ch") ("mpdan" "ph" "ps" "ch" ("a1i" "ps" "ph" "mpan2_1") "mpan2_2")) ;; An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.) (theorem "mp2an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mp2an_1" "ph") ("mp2an_2" "ps") ("mp2an_3" ("wi" ("wa" "ph" "ps") "ch"))) (for) "ch" ("ax_mp" "ps" "ch" "mp2an_2" ("mpan" "ph" "ps" "ch" "mp2an_1" "mp2an_3"))) ;; An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2010.) (theorem "mp4an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mp4an_1" "ph") ("mp4an_2" "ps") ("mp4an_3" "ch") ("mp4an_4" "th") ("mp4an_5" ("wi" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) "ta"))) (for) "ta" ("mp2an" ("wa" "ph" "ps") ("wa" "ch" "th") "ta" ("pm3_2i" "ph" "ps" "mp4an_1" "mp4an_2") ("pm3_2i" "ch" "th" "mp4an_3" "mp4an_4") "mp4an_5")) ;; A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (theorem "mpan2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpan2d_1" ("wi" "ph" "ch")) ("mpan2d_2" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" ("wi" "ps" "th")) ("mpid" "ph" "ps" "ch" "th" "mpan2d_1" ("expd" "ph" "ps" "ch" "th" "mpan2d_2"))) ;; A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) (theorem "mpand" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpand_1" ("wi" "ph" "ps")) ("mpand_2" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" ("wi" "ch" "th")) ("mpan2d" "ph" "ch" "ps" "th" "mpand_1" ("ancomsd" "ph" "ps" "ch" "th" "mpand_2"))) ;; An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.) (theorem "mpani" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpani_1" "ps") ("mpani_2" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" ("wi" "ch" "th")) ("mpand" "ph" "ps" "ch" "th" ("a1i" "ps" "ph" "mpani_1") "mpani_2")) ;; An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.) (theorem "mpan2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpan2i_1" "ch") ("mpan2i_2" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" ("wi" "ps" "th")) ("mpan2d" "ph" "ps" "ch" "th" ("a1i" "ch" "ph" "mpan2i_1") "mpan2i_2")) ;; An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.) (theorem "mp2ani" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mp2ani_1" "ps") ("mp2ani_2" "ch") ("mp2ani_3" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" "th") ("mpi" "ph" "ch" "th" "mp2ani_2" ("mpani" "ph" "ps" "ch" "th" "mp2ani_1" "mp2ani_3"))) ;; A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (theorem "mp2and" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mp2and_1" ("wi" "ph" "ps")) ("mp2and_2" ("wi" "ph" "ch")) ("mp2and_3" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" "th") ("mpd" "ph" "ch" "th" "mp2and_2" ("mpand" "ph" "ps" "ch" "th" "mp2and_1" "mp2and_3"))) ;; An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) (theorem "mpanl1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpanl1_1" "ph") ("mpanl1_2" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" "ps" "ch") "th") ("sylan" "ps" ("wa" "ph" "ps") "ch" "th" ("jctl" "ps" "ph" "mpanl1_1") "mpanl1_2")) ;; An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "mpanl2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpanl2_1" "ps") ("mpanl2_2" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" "ph" "ch") "th") ("sylan" "ph" ("wa" "ph" "ps") "ch" "th" ("jctr" "ph" "ps" "mpanl2_1") "mpanl2_2")) ;; An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.) (theorem "mpanl12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpanl12_1" "ph") ("mpanl12_2" "ps") ("mpanl12_3" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" "ch" "th") ("mpan" "ps" "ch" "th" "mpanl12_2" ("mpanl1" "ph" "ps" "ch" "th" "mpanl12_1" "mpanl12_3"))) ;; An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "mpanr1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpanr1_1" "ps") ("mpanr1_2" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ph" "ch") "th") ("mpanl2" "ph" "ps" "ch" "th" "mpanr1_1" ("anassrs" "ph" "ps" "ch" "th" "mpanr1_2"))) ;; An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) (theorem "mpanr2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpanr2_1" "ch") ("mpanr2_2" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ph" "ps") "th") ("sylan2" "ps" "ph" ("wa" "ps" "ch") "th" ("jctr" "ps" "ch" "mpanr2_1") "mpanr2_2")) ;; An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.) (theorem "mpanr12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpanr12_1" "ps") ("mpanr12_2" "ch") ("mpanr12_3" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" "ph" "th") ("mpan2" "ph" "ch" "th" "mpanr12_2" ("mpanr1" "ph" "ps" "ch" "th" "mpanr12_1" "mpanr12_3"))) ;; An inference based on modus ponens. (Contributed by NM, 30-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) (theorem "mpanlr1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mpanlr1_1" "ps") ("mpanlr1_2" ("wi" ("wa" ("wa" "ph" ("wa" "ps" "ch")) "th") "ta"))) (for) ("wi" ("wa" ("wa" "ph" "ch") "th") "ta") ("sylanl2" "ch" "ph" ("wa" "ps" "ch") "th" "ta" ("jctl" "ch" "ps" "mpanlr1_1") "mpanlr1_2")) ;; Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.) (theorem "pm5_74da" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm5_74da_1" ("wi" ("wa" "ph" "ps") ("wb" "ch" "th")))) (for) ("wi" "ph" ("wb" ("wi" "ps" "ch") ("wi" "ps" "th"))) ("pm5_74d" "ph" "ps" "ch" "th" ("ex" "ph" "ps" ("wb" "ch" "th") "pm5_74da_1"))) ;; Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_45" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" "ph" ("wa" "ph" ("wo" "ph" "ps"))) ("pm4_71i" "ph" ("wo" "ph" "ps") ("orc" "ph" "ps"))) ;; Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.) (theorem "imdistan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wi" "ps" "ch")) ("wi" ("wa" "ph" "ps") ("wa" "ph" "ch"))) ("bitr4i" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ph" ("wi" "ps" ("wa" "ph" "ch"))) ("wi" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("pm5_42" "ph" "ps" "ch") ("impexp" "ph" "ps" ("wa" "ph" "ch")))) ;; Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.) (theorem "imdistani" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("imdistani_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("imp" "ph" "ps" ("wa" "ph" "ch") ("anc2li" "ph" "ps" "ch" "imdistani_1"))) ;; Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.) (theorem "imdistanri" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("imdistanri_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" ("wa" "ps" "ph") ("wa" "ch" "ph")) ("impac" "ps" "ph" "ch" ("com12" "ph" "ps" "ch" "imdistanri_1"))) ;; Distribution of implication with conjunction (deduction rule). (Contributed by NM, 27-Aug-2004.) (theorem "imdistand" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("imdistand_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" "ph" ("wi" ("wa" "ps" "ch") ("wa" "ps" "th"))) ("sylib" "ph" ("wi" "ps" ("wi" "ch" "th")) ("wi" ("wa" "ps" "ch") ("wa" "ps" "th")) "imdistand_1" ("imdistan" "ps" "ch" "th"))) ;; Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.) (theorem "imdistanda" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("imdistanda_1" ("wi" ("wa" "ph" "ps") ("wi" "ch" "th")))) (for) ("wi" "ph" ("wi" ("wa" "ps" "ch") ("wa" "ps" "th"))) ("imdistand" "ph" "ps" "ch" "th" ("ex" "ph" "ps" ("wi" "ch" "th") "imdistanda_1"))) ;; Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.) (theorem "anbi2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anbi_1" ("wb" "ph" "ps"))) (for) ("wb" ("wa" "ch" "ph") ("wa" "ch" "ps")) ("pm5_32i" "ch" "ph" "ps" ("a1i" ("wb" "ph" "ps") "ch" "anbi_1"))) ;; Introduce a right conjunct to both sides of a logical equivalence. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.) (theorem "anbi1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anbi_1" ("wb" "ph" "ps"))) (for) ("wb" ("wa" "ph" "ch") ("wa" "ps" "ch")) ("pm5_32ri" "ch" "ph" "ps" ("a1i" ("wb" "ph" "ps") "ch" "anbi_1"))) ;; Variant of ~ anbi2i with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (theorem "anbi2ci" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anbi_1" ("wb" "ph" "ps"))) (for) ("wb" ("wa" "ph" "ch") ("wa" "ch" "ps")) ("bitri" ("wa" "ph" "ch") ("wa" "ps" "ch") ("wa" "ch" "ps") ("anbi1i" "ph" "ps" "ch" "anbi_1") ("ancom" "ps" "ch"))) ;; Conjoin both sides of two equivalences. (Contributed by NM, 12-Mar-1993.) (theorem "anbi12i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("anbi12_1" ("wb" "ph" "ps")) ("anbi12_2" ("wb" "ch" "th"))) (for) ("wb" ("wa" "ph" "ch") ("wa" "ps" "th")) ("bitri" ("wa" "ph" "ch") ("wa" "ps" "ch") ("wa" "ps" "th") ("anbi1i" "ph" "ps" "ch" "anbi12_1") ("anbi2i" "ch" "th" "ps" "anbi12_2"))) ;; Variant of ~ anbi12i with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (theorem "anbi12ci" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("anbi12_1" ("wb" "ph" "ps")) ("anbi12_2" ("wb" "ch" "th"))) (for) ("wb" ("wa" "ph" "ch") ("wa" "th" "ps")) ("bitri" ("wa" "ph" "ch") ("wa" "ps" "th") ("wa" "th" "ps") ("anbi12i" "ph" "ps" "ch" "th" "anbi12_1" "anbi12_2") ("ancom" "ps" "th"))) ;; A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.) (theorem "syldanl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syldanl_1" ("wi" ("wa" "ph" "ps") "ch")) ("syldanl_2" ("wi" ("wa" ("wa" "ph" "ch") "th") "ta"))) (for) ("wi" ("wa" ("wa" "ph" "ps") "th") "ta") ("sylan" ("wa" "ph" "ps") ("wa" "ph" "ch") "th" "ta" ("imdistani" "ph" "ps" "ch" ("ex" "ph" "ps" "ch" "syldanl_1")) "syldanl_2")) ;; Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.) (theorem "sylan9bb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylan9bb_1" ("wi" "ph" ("wb" "ps" "ch"))) ("sylan9bb_2" ("wi" "th" ("wb" "ch" "ta")))) (for) ("wi" ("wa" "ph" "th") ("wb" "ps" "ta")) ("bitrd" ("wa" "ph" "th") "ps" "ch" "ta" ("adantr" "ph" ("wb" "ps" "ch") "th" "sylan9bb_1") ("adantl" "th" ("wb" "ch" "ta") "ph" "sylan9bb_2"))) ;; Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.) (theorem "sylan9bbr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("sylan9bbr_1" ("wi" "ph" ("wb" "ps" "ch"))) ("sylan9bbr_2" ("wi" "th" ("wb" "ch" "ta")))) (for) ("wi" ("wa" "th" "ph") ("wb" "ps" "ta")) ("ancoms" "ph" "th" ("wb" "ps" "ta") ("sylan9bb" "ph" "ps" "ch" "th" "ta" "sylan9bbr_1" "sylan9bbr_2"))) ;; Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.) (theorem "orbi2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("bid_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("wo" "th" "ps") ("wo" "th" "ch"))) ("_3bitr4g" "ph" ("wi" ("wn" "th") "ps") ("wi" ("wn" "th") "ch") ("wo" "th" "ps") ("wo" "th" "ch") ("imbi2d" "ph" "ps" "ch" ("wn" "th") "bid_1") ("df_or" "th" "ps") ("df_or" "th" "ch"))) ;; Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.) (theorem "orbi1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("bid_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("wo" "ps" "th") ("wo" "ch" "th"))) ("_3bitr4g" "ph" ("wo" "th" "ps") ("wo" "th" "ch") ("wo" "ps" "th") ("wo" "ch" "th") ("orbi2d" "ph" "ps" "ch" "th" "bid_1") ("orcom" "ps" "th") ("orcom" "ch" "th"))) ;; Deduction adding a left conjunct to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.) (theorem "anbi2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("bid_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("wa" "th" "ps") ("wa" "th" "ch"))) ("pm5_32d" "ph" "th" "ps" "ch" ("a1d" "ph" ("wb" "ps" "ch") "th" "bid_1"))) ;; Deduction adding a right conjunct to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.) (theorem "anbi1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("bid_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("wa" "ps" "th") ("wa" "ch" "th"))) ("pm5_32rd" "ph" "th" "ps" "ch" ("a1d" "ph" ("wb" "ps" "ch") "th" "bid_1"))) ;; Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) (theorem "orbi1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wb" ("wo" "ph" "ch") ("wo" "ps" "ch"))) ("orbi1d" ("wb" "ph" "ps") "ph" "ps" "ch" ("id" ("wb" "ph" "ps")))) ;; Introduce a right conjunct to both sides of a logical equivalence. Theorem *4.36 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) (theorem "anbi1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wb" ("wa" "ph" "ch") ("wa" "ps" "ch"))) ("anbi1d" ("wb" "ph" "ps") "ph" "ps" "ch" ("id" ("wb" "ph" "ps")))) ;; Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.) (theorem "anbi2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wb" ("wa" "ch" "ph") ("wa" "ch" "ps"))) ("anbi2d" ("wb" "ph" "ps") "ph" "ps" "ch" ("id" ("wb" "ph" "ps")))) ;; Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (theorem "bitr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" ("wb" "ph" "ps") ("wb" "ps" "ch")) ("wb" "ph" "ch")) ("biimpar" ("wb" "ph" "ps") ("wb" "ph" "ch") ("wb" "ps" "ch") ("bibi1" "ph" "ps" "ch"))) ;; Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 21-Jun-1993.) (theorem "orbi12d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("bi12d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("bi12d_2" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" ("wo" "ps" "th") ("wo" "ch" "ta"))) ("bitrd" "ph" ("wo" "ps" "th") ("wo" "ch" "th") ("wo" "ch" "ta") ("orbi1d" "ph" "ps" "ch" "th" "bi12d_1") ("orbi2d" "ph" "th" "ta" "ch" "bi12d_2"))) ;; Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 26-May-1993.) (theorem "anbi12d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("bi12d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("bi12d_2" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" ("wa" "ps" "th") ("wa" "ch" "ta"))) ("bitrd" "ph" ("wa" "ps" "th") ("wa" "ch" "th") ("wa" "ch" "ta") ("anbi1d" "ph" "ps" "ch" "th" "bi12d_1") ("anbi2d" "ph" "th" "ta" "ch" "bi12d_2"))) ;; Theorem *5.3 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "pm5_3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" ("wa" "ph" "ps") "ch") ("wi" ("wa" "ph" "ps") ("wa" "ph" "ch"))) ("bitri" ("wi" ("wa" "ph" "ps") "ch") ("wi" "ph" ("wi" "ps" "ch")) ("wi" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("impexp" "ph" "ps" "ch") ("imdistan" "ph" "ps" "ch"))) ;; Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.) (theorem "pm5_61" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" ("wo" "ph" "ps") ("wn" "ps")) ("wa" "ph" ("wn" "ps"))) ("pm5_32ri" ("wn" "ps") ("wo" "ph" "ps") "ph" ("syl6rbb" ("wn" "ps") "ph" ("wo" "ps" "ph") ("wo" "ph" "ps") ("biorf" "ps" "ph") ("orcom" "ps" "ph")))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) (theorem "adantll" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("adant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" "th" "ph") "ps") "ch") ("sylan" ("wa" "th" "ph") "ph" "ps" "ch" ("simpr" "th" "ph") "adant2_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) (theorem "adantlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("adant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" "ph" "th") "ps") "ch") ("sylan" ("wa" "ph" "th") "ph" "ps" "ch" ("simpl" "ph" "th") "adant2_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) (theorem "adantrl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("adant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" "ph" ("wa" "th" "ps")) "ch") ("sylan2" ("wa" "th" "ps") "ph" "ps" "ch" ("simpr" "th" "ps") "adant2_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) (theorem "adantrr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("adant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" "ph" ("wa" "ps" "th")) "ch") ("sylan2" ("wa" "ps" "th") "ph" "ps" "ch" ("simpl" "ps" "th") "adant2_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 2-Dec-2012.) (theorem "adantlll" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("adantl2_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" "ta" "ph") "ps") "ch") "th") ("sylanl1" ("wa" "ta" "ph") "ph" "ps" "ch" "th" ("simpr" "ta" "ph") "adantl2_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (theorem "adantllr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("adantl2_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ta") "ps") "ch") "th") ("sylanl1" ("wa" "ph" "ta") "ph" "ps" "ch" "th" ("simpl" "ph" "ta") "adantl2_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (theorem "adantlrl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("adantl2_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" ("wa" "ph" ("wa" "ta" "ps")) "ch") "th") ("sylanl2" ("wa" "ta" "ps") "ph" "ps" "ch" "th" ("simpr" "ta" "ps") "adantl2_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (theorem "adantlrr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("adantl2_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" ("wa" "ph" ("wa" "ps" "ta")) "ch") "th") ("sylanl2" ("wa" "ps" "ta") "ph" "ps" "ch" "th" ("simpl" "ps" "ta") "adantl2_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (theorem "adantrll" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("adantr2_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ph" ("wa" ("wa" "ta" "ps") "ch")) "th") ("sylanr1" ("wa" "ta" "ps") "ph" "ps" "ch" "th" ("simpr" "ta" "ps") "adantr2_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (theorem "adantrlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("adantr2_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ph" ("wa" ("wa" "ps" "ta") "ch")) "th") ("sylanr1" ("wa" "ps" "ta") "ph" "ps" "ch" "th" ("simpl" "ps" "ta") "adantr2_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (theorem "adantrrl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("adantr2_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ph" ("wa" "ps" ("wa" "ta" "ch"))) "th") ("sylanr2" ("wa" "ta" "ch") "ph" "ps" "ch" "th" ("simpr" "ta" "ch") "adantr2_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (theorem "adantrrr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("adantr2_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ph" ("wa" "ps" ("wa" "ch" "ta"))) "th") ("sylanr2" ("wa" "ch" "ta") "ph" "ps" "ch" "th" ("simpl" "ch" "ta") "adantr2_1")) ;; Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.) (theorem "ad2antrr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" "ph" "ch") "th") "ps") ("adantlr" "ph" "th" "ps" "ch" ("adantr" "ph" "ps" "th" "ad2ant_1"))) ;; Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.) (theorem "ad2antlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" "ch" "ph") "th") "ps") ("adantll" "ph" "th" "ps" "ch" ("adantr" "ph" "ps" "th" "ad2ant_1"))) ;; Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (theorem "ad2antrl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" "ch" ("wa" "ph" "th")) "ps") ("adantl" ("wa" "ph" "th") "ps" "ch" ("adantr" "ph" "ps" "th" "ad2ant_1"))) ;; Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (theorem "ad2antll" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" "ch" ("wa" "th" "ph")) "ps") ("adantl" ("wa" "th" "ph") "ps" "ch" ("adantl" "ph" "ps" "th" "ad2ant_1"))) ;; Deduction adding three conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.) (theorem "ad3antrrr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "ps") ("ad2antrr" ("wa" "ph" "ch") "ps" "th" "ta" ("adantr" "ph" "ps" "ch" "ad2ant_1"))) ;; Deduction adding three conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad3antlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "ps") ("ad2antrr" ("wa" "ch" "ph") "ps" "th" "ta" ("adantl" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad3antlr as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad3antlrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "ps") ("adantr" ("wa" ("wa" "ch" "ph") "th") "ps" "ta" ("ad2antlr" "ph" "ps" "ch" "th" "ad2ant_1"))) ;; Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad4antr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ps") ("ad3antrrr" ("wa" "ph" "ch") "ps" "th" "ta" "et" ("adantr" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad4antr as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad4antrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ps") ("adantr" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "ps" "et" ("ad3antrrr" "ph" "ps" "ch" "th" "ta" "ad2ant_1"))) ;; Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad4antlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ps") ("ad3antrrr" ("wa" "ch" "ph") "ps" "th" "ta" "et" ("adantl" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad4antlr as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad4antlrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ps") ("adantr" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "ps" "et" ("ad3antlr" "ph" "ps" "ch" "th" "ta" "ad2ant_1"))) ;; Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad5antr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "ps") ("ad4antr" ("wa" "ph" "ch") "ps" "th" "ta" "et" "ze" ("adantr" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad5antr as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5antrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ps" "ze" ("ad4antr" "ph" "ps" "ch" "th" "ta" "et" "ad2ant_1"))) ;; Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad5antlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "ps") ("ad4antr" ("wa" "ch" "ph") "ps" "th" "ta" "et" "ze" ("adantl" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad5antlr as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5antlrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ps" "ze" ("ad4antlr" "ph" "ps" "ch" "th" "ta" "et" "ad2ant_1"))) ;; Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad6antr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "ps") ("ad5antr" ("wa" "ph" "ch") "ps" "th" "ta" "et" "ze" "si" ("adantr" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad6antr as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad6antrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "ps" "si" ("ad5antr" "ph" "ps" "ch" "th" "ta" "et" "ze" "ad2ant_1"))) ;; Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad6antlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "ps") ("ad5antr" ("wa" "ch" "ph") "ps" "th" "ta" "et" "ze" "si" ("adantl" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad6antlr as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad6antlrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "ps" "si" ("ad5antlr" "ph" "ps" "ch" "th" "ta" "et" "ze" "ad2ant_1"))) ;; Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad7antr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "rh") "ps") ("ad6antr" ("wa" "ph" "ch") "ps" "th" "ta" "et" "ze" "si" "rh" ("adantr" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad7antr as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad7antrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "rh") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "ps" "rh" ("ad6antr" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "ad2ant_1"))) ;; Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad7antlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "rh") "ps") ("ad6antr" ("wa" "ch" "ph") "ps" "th" "ta" "et" "ze" "si" "rh" ("adantl" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad7antlr as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad7antlrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "rh") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "ps" "rh" ("ad6antlr" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "ad2ant_1"))) ;; Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad8antr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "ps") ("ad7antr" ("wa" "ph" "ch") "ps" "th" "ta" "et" "ze" "si" "rh" "mu" ("adantr" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad8antr as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad8antrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "rh") "ps" "mu" ("ad7antr" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "ad2ant_1"))) ;; Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad8antlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "rh") "mu") "ps") ("ad7antr" ("wa" "ch" "ph") "ps" "th" "ta" "et" "ze" "si" "rh" "mu" ("adantl" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad8antlr as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad8antlrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "rh") "mu") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "rh") "ps" "mu" ("ad7antlr" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "ad2ant_1"))) ;; Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad9antr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ps") ("ad8antr" ("wa" "ph" "ch") "ps" "th" "ta" "et" "ze" "si" "rh" "mu" "la" ("adantr" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad9antr as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad9antrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "ps" "la" ("ad8antr" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" "ad2ant_1"))) ;; Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad9antlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ps") ("ad8antr" ("wa" "ch" "ph") "ps" "th" "ta" "et" "ze" "si" "rh" "mu" "la" ("adantl" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad9antlr as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad9antlrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "rh") "mu") "ps" "la" ("ad8antlr" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" "ad2ant_1"))) ;; Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad10antr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff")) ("ka" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ka") "ps") ("ad9antr" ("wa" "ph" "ch") "ps" "th" "ta" "et" "ze" "si" "rh" "mu" "la" "ka" ("adantr" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad10antr as of 5-Apr-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad10antrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff")) ("ka" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ka") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ps" "ka" ("ad9antr" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" "la" "ad2ant_1"))) ;; Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof shortened by Wolf Lammen, 5-Apr-2022.) (theorem "ad10antlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff")) ("ka" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ka") "ps") ("ad9antr" ("wa" "ch" "ph") "ps" "th" "ta" "et" "ze" "si" "rh" "mu" "la" "ka" ("adantl" "ph" "ps" "ch" "ad2ant_1"))) ;; Obsolete version of ~ ad10antlr as of 5-Apr-2022. (Contributed by Mario Carneiro, 5-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad10antlrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff")) ("ka" ( "wff"))) (for ("ad2ant_1" ("wi" "ph" "ps"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ka") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ch" "ph") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ps" "ka" ("ad9antlr" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" "la" "ad2ant_1"))) ;; Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.) (theorem "ad2ant2l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad2ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" "th" "ph") ("wa" "ta" "ps")) "ch") ("adantll" "ph" ("wa" "ta" "ps") "ch" "th" ("adantrl" "ph" "ps" "ch" "ta" "ad2ant2_1"))) ;; Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.) (theorem "ad2ant2r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad2ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" "ph" "th") ("wa" "ps" "ta")) "ch") ("adantlr" "ph" ("wa" "ps" "ta") "ch" "th" ("adantrr" "ph" "ps" "ch" "ta" "ad2ant2_1"))) ;; Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.) (theorem "ad2ant2lr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad2ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" "th" "ph") ("wa" "ps" "ta")) "ch") ("adantll" "ph" ("wa" "ps" "ta") "ch" "th" ("adantrr" "ph" "ps" "ch" "ta" "ad2ant2_1"))) ;; Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.) (theorem "ad2ant2rl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad2ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" "ph" "th") ("wa" "ta" "ps")) "ch") ("adantlr" "ph" ("wa" "ta" "ps") "ch" "th" ("adantrl" "ph" "ps" "ch" "ta" "ad2ant2_1"))) ;; Deduction adding 1 conjunct to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (theorem "adantl3r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("adantl3r_1" ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "et") "ps") "ch") "th") "ta") ("imp" ("wa" ("wa" ("wa" "ph" "et") "ps") "ch") "th" "ta" ("adantllr" "ph" "ps" "ch" ("wi" "th" "ta") "et" ("ex" ("wa" ("wa" "ph" "ps") "ch") "th" "ta" "adantl3r_1")))) ;; Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.) (theorem "adantl4r" (for ("ph" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff")) ("ka" ( "wff"))) (for ("adantl4r_1" ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "si") "rh") "mu") "la") "ka"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ze") "si") "rh") "mu") "la") "ka") ("imp" ("wa" ("wa" ("wa" ("wa" "ph" "ze") "si") "rh") "mu") "la" "ka" ("adantl3r" "ph" "si" "rh" "mu" ("wi" "la" "ka") "ze" ("ex" ("wa" ("wa" ("wa" "ph" "si") "rh") "mu") "la" "ka" "adantl4r_1")))) ;; Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.) (theorem "adantl5r" (for ("ph" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff")) ("ka" ( "wff"))) (for ("adantl5r_1" ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ze") "si") "rh") "mu") "la") "ka"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "et") "ze") "si") "rh") "mu") "la") "ka") ("imp" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "et") "ze") "si") "rh") "mu") "la" "ka" ("adantl4r" "ph" "et" "ze" "si" "rh" "mu" ("wi" "la" "ka") ("ex" ("wa" ("wa" ("wa" ("wa" "ph" "ze") "si") "rh") "mu") "la" "ka" "adantl5r_1")))) ;; Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.) (theorem "adantl6r" (for ("ph" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff")) ("ka" ( "wff"))) (for ("adantl6r_1" ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "et") "ze") "si") "rh") "mu") "la") "ka"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ta") "et") "ze") "si") "rh") "mu") "la") "ka") ("imp" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ta") "et") "ze") "si") "rh") "mu") "la" "ka" ("adantl5r" "ph" "ta" "et" "ze" "si" "rh" "mu" ("wi" "la" "ka") ("ex" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "et") "ze") "si") "rh") "mu") "la" "ka" "adantl6r_1")))) ;; Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.) (theorem "simpll" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" "ph" "ps") "ch") "ph") ("ad2antrr" "ph" "ph" "ps" "ch" ("id" "ph"))) ;; Deduction form of ~ simpll , eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (theorem "simplld" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("simplld_1" ("wi" "ph" ("wa" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" "ps") ("simpld" "ph" "ps" "ch" ("simpld" "ph" ("wa" "ps" "ch") "th" "simplld_1"))) ;; Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.) (theorem "simplr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" "ph" "ps") "ch") "ps") ("ad2antlr" "ps" "ps" "ph" "ch" ("id" "ps"))) ;; Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (theorem "simplrd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("simplrd_1" ("wi" "ph" ("wa" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" "ch") ("simprd" "ph" "ps" "ch" ("simpld" "ph" ("wa" "ps" "ch") "th" "simplrd_1"))) ;; Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.) (theorem "simprl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("wa" "ps" "ch")) "ps") ("ad2antrl" "ps" "ps" "ph" "ch" ("id" "ps"))) ;; Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (theorem "simprld" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("simprld_1" ("wi" "ph" ("wa" "ps" ("wa" "ch" "th"))))) (for) ("wi" "ph" "ch") ("simpld" "ph" "ch" "th" ("simprd" "ph" "ps" ("wa" "ch" "th") "simprld_1"))) ;; Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.) (theorem "simprr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("wa" "ps" "ch")) "ch") ("ad2antll" "ch" "ch" "ph" "ps" ("id" "ch"))) ;; Deduction form of ~ simprr , eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (theorem "simprrd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("simprrd_1" ("wi" "ph" ("wa" "ps" ("wa" "ch" "th"))))) (for) ("wi" "ph" "th") ("simprd" "ph" "ch" "th" ("simprd" "ph" "ps" ("wa" "ch" "th") "simprrd_1"))) ;; Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (Proof shortened by Wolf Lammen, 6-Apr-2022.) (theorem "simplll" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ph") ("ad3antrrr" "ph" "ph" "ps" "ch" "th" ("id" "ph"))) ;; Obsolete version of ~ simplll as of 6-Apr-2022. (Contributed by Jeff Hankins, 28-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simplllOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ph") ("ad2antrr" ("wa" "ph" "ps") "ph" "ch" "th" ("simpl" "ph" "ps"))) ;; Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (Proof shortened by Wolf Lammen, 6-Apr-2022.) (theorem "simpllr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ps") ("ad3antlr" "ps" "ps" "ph" "ch" "th" ("id" "ps"))) ;; Obsolete version of ~ simpllr as of 6-Apr-2022. (Contributed by Jeff Hankins, 28-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpllrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ps") ("ad2antrr" ("wa" "ph" "ps") "ps" "ch" "th" ("simpr" "ph" "ps"))) ;; Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (theorem "simplrl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" "ph" ("wa" "ps" "ch")) "th") "ps") ("ad2antlr" ("wa" "ps" "ch") "ps" "ph" "th" ("simpl" "ps" "ch"))) ;; Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (theorem "simplrr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" "ph" ("wa" "ps" "ch")) "th") "ch") ("ad2antlr" ("wa" "ps" "ch") "ch" "ph" "th" ("simpr" "ps" "ch"))) ;; Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (theorem "simprll" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("wa" ("wa" "ps" "ch") "th")) "ps") ("ad2antrl" ("wa" "ps" "ch") "ps" "ph" "th" ("simpl" "ps" "ch"))) ;; Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (theorem "simprlr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("wa" ("wa" "ps" "ch") "th")) "ch") ("ad2antrl" ("wa" "ps" "ch") "ch" "ph" "th" ("simpr" "ps" "ch"))) ;; Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (theorem "simprrl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("wa" "ps" ("wa" "ch" "th"))) "ch") ("ad2antll" ("wa" "ch" "th") "ch" "ph" "ps" ("simpl" "ch" "th"))) ;; Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.) (theorem "simprrr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("wa" "ps" ("wa" "ch" "th"))) "th") ("ad2antll" ("wa" "ch" "th") "th" "ph" "ps" ("simpr" "ch" "th"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_4l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "ph") ("ad4antr" "ph" "ph" "ps" "ch" "th" "ta" ("id" "ph"))) ;; Obsolete proof of ~ simp-4l as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_4lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "ph") ("adantr" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ph" "ta" ("simplll" "ph" "ps" "ch" "th"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_4r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "ps") ("ad3antrrr" ("wa" "ph" "ps") "ps" "ch" "th" "ta" ("simpr" "ph" "ps"))) ;; Obsolete proof of ~ simp-4r as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_4rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "ps") ("adantr" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ps" "ta" ("simpllr" "ph" "ps" "ch" "th"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_5l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ph") ("ad5antr" "ph" "ph" "ps" "ch" "th" "ta" "et" ("id" "ph"))) ;; Obsolete proof of ~ simp-5l as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_5lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ph") ("adantr" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "ph" "et" ("simp_4l" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_5r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ps") ("ad4antr" ("wa" "ph" "ps") "ps" "ch" "th" "ta" "et" ("simpr" "ph" "ps"))) ;; Obsolete proof of ~ simp-5r as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_5rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "ps" "et" ("simp_4r" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_6l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "ph") ("ad6antr" "ph" "ph" "ps" "ch" "th" "ta" "et" "ze" ("id" "ph"))) ;; Obsolete proof of ~ simp-6l as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_6lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "ph") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ph" "ze" ("simp_5l" "ph" "ps" "ch" "th" "ta" "et"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_6r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "ps") ("ad5antr" ("wa" "ph" "ps") "ps" "ch" "th" "ta" "et" "ze" ("simpr" "ph" "ps"))) ;; Obsolete proof of ~ simp-6r as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_6rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ps" "ze" ("simp_5r" "ph" "ps" "ch" "th" "ta" "et"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_7l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "ph") ("ad7antr" "ph" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" ("id" "ph"))) ;; Obsolete proof of ~ simp-7l as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_7lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "ph") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "ph" "si" ("simp_6l" "ph" "ps" "ch" "th" "ta" "et" "ze"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_7r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "ps") ("ad6antr" ("wa" "ph" "ps") "ps" "ch" "th" "ta" "et" "ze" "si" ("simpr" "ph" "ps"))) ;; Obsolete proof of ~ simp-7r as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_7rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "ps" "si" ("simp_6r" "ph" "ps" "ch" "th" "ta" "et" "ze"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_8l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "ph") ("ad8antr" "ph" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" ("id" "ph"))) ;; Obsolete proof of ~ simp-8l as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_8lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "ph") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "ph" "rh" ("simp_7l" "ph" "ps" "ch" "th" "ta" "et" "ze" "si"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_8r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "ps") ("ad7antr" ("wa" "ph" "ps") "ps" "ch" "th" "ta" "et" "ze" "si" "rh" ("simpr" "ph" "ps"))) ;; Obsolete proof of ~ simp-8r as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_8rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "ps" "rh" ("simp_7r" "ph" "ps" "ch" "th" "ta" "et" "ze" "si"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_9l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "ph") ("ad9antr" "ph" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" ("id" "ph"))) ;; Obsolete proof of ~ simp-9l as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_9lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "ph") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "ph" "mu" ("simp_8l" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_9r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "ps") ("ad8antr" ("wa" "ph" "ps") "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" ("simpr" "ph" "ps"))) ;; Obsolete proof of ~ simp-9r as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_9rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "ps" "mu" ("simp_8r" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_10l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ph") ("ad10antr" "ph" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" "la" ("id" "ph"))) ;; Obsolete proof of ~ simp-10l as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_10lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ph") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "ph" "la" ("simp_9l" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_10r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ps") ("ad9antr" ("wa" "ph" "ps") "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" "la" ("simpr" "ph" "ps"))) ;; Obsolete proof of ~ simp-10r as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_10rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "ps" "la" ("simp_9r" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_11l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff")) ("ka" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ka") "ph") ("ad10antr" ("wa" "ph" "ps") "ph" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" "la" "ka" ("simpl" "ph" "ps"))) ;; Obsolete proof of ~ simp-11l as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_11lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff")) ("ka" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ka") "ph") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ph" "ka" ("simp_10l" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" "la"))) ;; Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.) (theorem "simp_11r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff")) ("ka" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ka") "ps") ("ad10antr" ("wa" "ph" "ps") "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" "la" "ka" ("simpr" "ph" "ps"))) ;; Obsolete proof of ~ simp-11r as of 24-May-2022. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp_11rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff")) ("ka" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ka") "ps") ("adantr" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") "et") "ze") "si") "rh") "mu") "la") "ps" "ka" ("simp_10r" "ph" "ps" "ch" "th" "ta" "et" "ze" "si" "rh" "mu" "la"))) ;; Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 9-Dec-2012.) (theorem "jaob" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" ("wo" "ph" "ch") "ps") ("wa" ("wi" "ph" "ps") ("wi" "ch" "ps"))) ("impbii" ("wi" ("wo" "ph" "ch") "ps") ("wa" ("wi" "ph" "ps") ("wi" "ch" "ps")) ("jca" ("wi" ("wo" "ph" "ch") "ps") ("wi" "ph" "ps") ("wi" "ch" "ps") ("pm2_67_2" "ph" "ps" "ch") ("imim1i" "ch" ("wo" "ph" "ch") "ps" ("olc" "ch" "ph"))) ("pm3_44" "ps" "ph" "ch"))) ;; Obsolete as of 2-Oct-2021. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.) (theorem "adant423OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("adant423_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "th") "ta") "ps") "ch") ("sylancom" ("wa" ("wa" "ph" "th") "ta") "ps" "ph" "ch" ("simplll" "ph" "th" "ta" "ps") "adant423_1")) ;; Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.) (theorem "jaoian" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jaoian_1" ("wi" ("wa" "ph" "ps") "ch")) ("jaoian_2" ("wi" ("wa" "th" "ps") "ch"))) (for) ("wi" ("wa" ("wo" "ph" "th") "ps") "ch") ("imp" ("wo" "ph" "th") "ps" "ch" ("jaoi" "ph" ("wi" "ps" "ch") "th" ("ex" "ph" "ps" "ch" "jaoian_1") ("ex" "th" "ps" "ch" "jaoian_2")))) ;; Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.) (theorem "jao1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("jao1i_1" ("wi" "ps" ("wi" "ch" "ph")))) (for) ("wi" ("wo" "ph" "ps") ("wi" "ch" "ph")) ("jaoi" "ph" ("wi" "ch" "ph") "ps" ("ax_1" "ph" "ch") "jao1i_1")) ;; Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.) (theorem "jaodan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jaodan_1" ("wi" ("wa" "ph" "ps") "ch")) ("jaodan_2" ("wi" ("wa" "ph" "th") "ch"))) (for) ("wi" ("wa" "ph" ("wo" "ps" "th")) "ch") ("imp" "ph" ("wo" "ps" "th") "ch" ("jaod" "ph" "ps" "ch" "th" ("ex" "ph" "ps" "ch" "jaodan_1") ("ex" "ph" "th" "ch" "jaodan_2")))) ;; Eliminate a disjunction in a deduction. A translation of natural deduction rule ` \/ ` E ( ` \/ ` elimination), see ~ natded . (Contributed by Mario Carneiro, 29-May-2016.) (theorem "mpjaodan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("jaodan_1" ("wi" ("wa" "ph" "ps") "ch")) ("jaodan_2" ("wi" ("wa" "ph" "th") "ch")) ("jaodan_3" ("wi" "ph" ("wo" "ps" "th")))) (for) ("wi" "ph" "ch") ("mpdan" "ph" ("wo" "ps" "th") "ch" "jaodan_3" ("jaodan" "ph" "ps" "ch" "th" "jaodan_1" "jaodan_2"))) ;; Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_77" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" ("wi" "ps" "ph") ("wi" "ch" "ph")) ("wi" ("wo" "ps" "ch") "ph")) ("bicomi" ("wi" ("wo" "ps" "ch") "ph") ("wa" ("wi" "ps" "ph") ("wi" "ch" "ph")) ("jaob" "ps" "ph" "ch"))) ;; Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_63" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wo" "ph" "ps") ("wi" ("wo" ("wn" "ph") "ps") "ps")) ("jaod" ("wo" "ph" "ps") ("wn" "ph") "ps" "ps" ("pm2_53" "ph" "ps") ("idd" ("wo" "ph" "ps") "ps"))) ;; Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_64" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wo" "ph" "ps") ("wi" ("wo" "ph" ("wn" "ps")) "ph")) ("com12" ("wo" "ph" ("wn" "ps")) ("wo" "ph" "ps") "ph" ("jao1i" "ph" ("wn" "ps") ("wo" "ph" "ps") ("orel2" "ps" "ph")))) ;; Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.) (theorem "pm2_61ian" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_61ian_1" ("wi" ("wa" "ph" "ps") "ch")) ("pm2_61ian_2" ("wi" ("wa" ("wn" "ph") "ps") "ch"))) (for) ("wi" "ps" "ch") ("pm2_61i" "ph" ("wi" "ps" "ch") ("ex" "ph" "ps" "ch" "pm2_61ian_1") ("ex" ("wn" "ph") "ps" "ch" "pm2_61ian_2"))) ;; Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.) (theorem "pm2_61dan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("pm2_61dan_1" ("wi" ("wa" "ph" "ps") "ch")) ("pm2_61dan_2" ("wi" ("wa" "ph" ("wn" "ps")) "ch"))) (for) ("wi" "ph" "ch") ("pm2_61d" "ph" "ps" "ch" ("ex" "ph" "ps" "ch" "pm2_61dan_1") ("ex" "ph" ("wn" "ps") "ch" "pm2_61dan_2"))) ;; Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.) (theorem "pm2_61ddan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm2_61ddan_1" ("wi" ("wa" "ph" "ps") "th")) ("pm2_61ddan_2" ("wi" ("wa" "ph" "ch") "th")) ("pm2_61ddan_3" ("wi" ("wa" "ph" ("wa" ("wn" "ps") ("wn" "ch"))) "th"))) (for) ("wi" "ph" "th") ("pm2_61dan" "ph" "ps" "th" "pm2_61ddan_1" ("pm2_61dan" ("wa" "ph" ("wn" "ps")) "ch" "th" ("adantlr" "ph" "ch" "th" ("wn" "ps") "pm2_61ddan_2") ("anassrs" "ph" ("wn" "ps") ("wn" "ch") "th" "pm2_61ddan_3")))) ;; Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.) (theorem "pm2_61dda" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm2_61dda_1" ("wi" ("wa" "ph" ("wn" "ps")) "th")) ("pm2_61dda_2" ("wi" ("wa" "ph" ("wn" "ch")) "th")) ("pm2_61dda_3" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" "ph" "th") ("pm2_61dan" "ph" "ps" "th" ("pm2_61dan" ("wa" "ph" "ps") "ch" "th" ("anassrs" "ph" "ps" "ch" "th" "pm2_61dda_3") ("adantlr" "ph" ("wn" "ch") "th" "ps" "pm2_61dda_2")) "pm2_61dda_1")) ;; Proof by contradiction. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Wolf Lammen, 19-Jun-2014.) (theorem "condan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("condan_1" ("wi" ("wa" "ph" ("wn" "ps")) "ch")) ("condan_2" ("wi" ("wa" "ph" ("wn" "ps")) ("wn" "ch")))) (for) ("wi" "ph" "ps") ("notnotrd" "ph" "ps" ("pm2_65da" "ph" ("wn" "ps") "ch" "condan_1" "condan_2"))) ;; Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Dec-2012.) (theorem "abai" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" "ph" "ps") ("wa" "ph" ("wi" "ph" "ps"))) ("pm5_32i" "ph" "ps" ("wi" "ph" "ps") ("biimt" "ph" "ps"))) ;; Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_53" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wi" ("wo" ("wo" "ph" "ps") "ch") "th") ("wa" ("wa" ("wi" "ph" "th") ("wi" "ps" "th")) ("wi" "ch" "th"))) ("bitri" ("wi" ("wo" ("wo" "ph" "ps") "ch") "th") ("wa" ("wi" ("wo" "ph" "ps") "th") ("wi" "ch" "th")) ("wa" ("wa" ("wi" "ph" "th") ("wi" "ps" "th")) ("wi" "ch" "th")) ("jaob" ("wo" "ph" "ps") "th" "ch") ("anbi1i" ("wi" ("wo" "ph" "ps") "th") ("wa" ("wi" "ph" "th") ("wi" "ps" "th")) ("wi" "ch" "th") ("jaob" "ph" "th" "ps")))) ;; Swap two conjuncts. Note that the first digit (1) in the label refers to the outer conjunct position, and the next digit (2) to the inner conjunct position. (Contributed by NM, 12-Mar-1995.) (theorem "an12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wa" "ps" "ch")) ("wa" "ps" ("wa" "ph" "ch"))) ("_3bitr3i" ("wa" ("wa" "ph" "ps") "ch") ("wa" ("wa" "ps" "ph") "ch") ("wa" "ph" ("wa" "ps" "ch")) ("wa" "ps" ("wa" "ph" "ch")) ("anbi1i" ("wa" "ph" "ps") ("wa" "ps" "ph") "ch" ("ancom" "ph" "ps")) ("anass" "ph" "ps" "ch") ("anass" "ps" "ph" "ch"))) ;; A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Wolf Lammen, 25-Dec-2012.) (theorem "an32" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" ("wa" "ph" "ps") "ch") ("wa" ("wa" "ph" "ch") "ps")) ("_3bitri" ("wa" ("wa" "ph" "ps") "ch") ("wa" "ph" ("wa" "ps" "ch")) ("wa" "ps" ("wa" "ph" "ch")) ("wa" ("wa" "ph" "ch") "ps") ("anass" "ph" "ps" "ch") ("an12" "ph" "ps" "ch") ("ancom" "ps" ("wa" "ph" "ch")))) ;; A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.) (theorem "an13" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wa" "ps" "ch")) ("wa" "ch" ("wa" "ps" "ph"))) ("_3bitr2i" ("wa" "ph" ("wa" "ps" "ch")) ("wa" "ps" ("wa" "ph" "ch")) ("wa" ("wa" "ps" "ph") "ch") ("wa" "ch" ("wa" "ps" "ph")) ("an12" "ph" "ps" "ch") ("anass" "ps" "ph" "ch") ("ancom" ("wa" "ps" "ph") "ch"))) ;; A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.) (theorem "an31" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" ("wa" "ph" "ps") "ch") ("wa" ("wa" "ch" "ps") "ph")) ("_3bitr4i" ("wa" "ph" ("wa" "ps" "ch")) ("wa" "ch" ("wa" "ps" "ph")) ("wa" ("wa" "ph" "ps") "ch") ("wa" ("wa" "ch" "ps") "ph") ("an13" "ph" "ps" "ch") ("anass" "ph" "ps" "ch") ("anass" "ch" "ps" "ph"))) ;; An inference to merge two lists of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.) (theorem "bianass" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("et" ( "wff"))) (for ("bianass_1" ("wb" "ph" ("wa" "ps" "ch")))) (for) ("wb" ("wa" "et" "ph") ("wa" ("wa" "et" "ps") "ch")) ("bitr4i" ("wa" "et" "ph") ("wa" "et" ("wa" "ps" "ch")) ("wa" ("wa" "et" "ps") "ch") ("anbi2i" "ph" ("wa" "ps" "ch") "et" "bianass_1") ("anass" "et" "ps" "ch"))) ;; Swap two conjuncts in antecedent. The label suffix "s" means that ~ an12 is combined with ~ syl (or a variant). (Contributed by NM, 13-Mar-1996.) (theorem "an12s" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("an12s_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ps" ("wa" "ph" "ch")) "th") ("sylbi" ("wa" "ps" ("wa" "ph" "ch")) ("wa" "ph" ("wa" "ps" "ch")) "th" ("an12" "ps" "ph" "ch") "an12s_1")) ;; Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) (theorem "ancom2s" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("an12s_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ph" ("wa" "ch" "ps")) "th") ("sylan2" ("wa" "ch" "ps") "ph" ("wa" "ps" "ch") "th" ("pm3_22" "ch" "ps") "an12s_1")) ;; Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.) (theorem "an13s" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("an12s_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ch" ("wa" "ps" "ph")) "th") ("imp32" "ch" "ps" "ph" "th" ("com13" "ph" "ps" "ch" "th" ("exp32" "ph" "ps" "ch" "th" "an12s_1")))) ;; Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.) (theorem "an32s" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("an32s_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" ("wa" "ph" "ch") "ps") "th") ("sylbi" ("wa" ("wa" "ph" "ch") "ps") ("wa" ("wa" "ph" "ps") "ch") "th" ("an32" "ph" "ch" "ps") "an32s_1")) ;; Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) (theorem "ancom1s" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("an32s_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" ("wa" "ps" "ph") "ch") "th") ("sylan" ("wa" "ps" "ph") ("wa" "ph" "ps") "ch" "th" ("pm3_22" "ps" "ph") "an32s_1")) ;; Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.) (theorem "an31s" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("an32s_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" ("wa" "ch" "ps") "ph") "th") ("imp31" "ch" "ps" "ph" "th" ("com13" "ph" "ps" "ch" "th" ("exp31" "ph" "ps" "ch" "th" "an32s_1")))) ;; Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.) (theorem "anass1rs" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("anass1rs_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" ("wa" "ph" "ch") "ps") "th") ("an32s" "ph" "ps" "ch" "th" ("anassrs" "ph" "ps" "ch" "th" "anass1rs_1"))) ;; Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.) (theorem "anabs1" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" ("wa" "ph" "ps") "ph") ("wa" "ph" "ps")) ("bicomi" ("wa" "ph" "ps") ("wa" ("wa" "ph" "ps") "ph") ("pm4_71i" ("wa" "ph" "ps") "ph" ("simpl" "ph" "ps")))) ;; Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.) (theorem "anabs5" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wa" "ph" "ps")) ("wa" "ph" "ps")) ("pm5_32i" "ph" ("wa" "ph" "ps") "ps" ("bicomd" "ph" "ps" ("wa" "ph" "ps") ("ibar" "ph" "ps")))) ;; Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.) (theorem "anabs7" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" "ps" ("wa" "ph" "ps")) ("wa" "ph" "ps")) ("bicomi" ("wa" "ph" "ps") ("wa" "ps" ("wa" "ph" "ps")) ("pm4_71ri" ("wa" "ph" "ps") "ps" ("simpr" "ph" "ps")))) ;; Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.) (theorem "a2and" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("rh" ( "wff"))) (for ("a2and_1" ("wi" "ph" ("wi" ("wa" "ps" "rh") ("wi" "ta" "th")))) ("a2and_2" ("wi" "ph" ("wi" ("wa" "ps" "rh") "ch")))) (for) ("wi" "ph" ("wi" ("wi" ("wa" "ps" "ch") "ta") ("wi" ("wa" "ps" "rh") "th"))) ("com23" "ph" ("wa" "ps" "rh") ("wi" ("wa" "ps" "ch") "ta") "th" ("ex" "ph" ("wa" "ps" "rh") ("wi" ("wi" ("wa" "ps" "ch") "ta") "th") ("embantd" ("wa" "ph" ("wa" "ps" "rh")) ("wa" "ps" "ch") "ta" "th" ("imp" "ph" ("wa" "ps" "rh") ("wa" "ps" "ch") ("imdistand" "ph" "ps" "rh" "ch" ("expd" "ph" "ps" "rh" "ch" "a2and_2"))) ("imp" "ph" ("wa" "ps" "rh") ("wi" "ta" "th") "a2and_1"))))) ;; Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (theorem "anabsan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anabsan_1" ("wi" ("wa" ("wa" "ph" "ph") "ps") "ch"))) (for) ("wi" ("wa" "ph" "ps") "ch") ("sylanb" "ph" ("wa" "ph" "ph") "ps" "ch" ("pm4_24" "ph") "anabsan_1")) ;; Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.) (theorem "anabss1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anabss1_1" ("wi" ("wa" ("wa" "ph" "ps") "ph") "ch"))) (for) ("wi" ("wa" "ph" "ps") "ch") ("anabsan" "ph" "ps" "ch" ("an32s" "ph" "ps" "ph" "ch" "anabss1_1"))) ;; Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (theorem "anabss4" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anabss4_1" ("wi" ("wa" ("wa" "ps" "ph") "ps") "ch"))) (for) ("wi" ("wa" "ph" "ps") "ch") ("ancoms" "ps" "ph" "ch" ("anabss1" "ps" "ph" "ch" "anabss4_1"))) ;; Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.) (theorem "anabss5" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anabss5_1" ("wi" ("wa" "ph" ("wa" "ph" "ps")) "ch"))) (for) ("wi" ("wa" "ph" "ps") "ch") ("anabsan" "ph" "ps" "ch" ("anassrs" "ph" "ph" "ps" "ch" "anabss5_1"))) ;; Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.) (theorem "anabsi5" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anabsi5_1" ("wi" "ph" ("wi" ("wa" "ph" "ps") "ch")))) (for) ("wi" ("wa" "ph" "ps") "ch") ("anabss5" "ph" "ps" "ch" ("imp" "ph" ("wa" "ph" "ps") "ch" "anabsi5_1"))) ;; Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.) (theorem "anabsi6" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anabsi6_1" ("wi" "ph" ("wi" ("wa" "ps" "ph") "ch")))) (for) ("wi" ("wa" "ph" "ps") "ch") ("anabsi5" "ph" "ps" "ch" ("ancomsd" "ph" "ps" "ph" "ch" "anabsi6_1"))) ;; Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.) (theorem "anabsi7" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anabsi7_1" ("wi" "ps" ("wi" ("wa" "ph" "ps") "ch")))) (for) ("wi" ("wa" "ph" "ps") "ch") ("ancoms" "ps" "ph" "ch" ("anabsi6" "ps" "ph" "ch" "anabsi7_1"))) ;; Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.) (theorem "anabsi8" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anabsi8_1" ("wi" "ps" ("wi" ("wa" "ps" "ph") "ch")))) (for) ("wi" ("wa" "ph" "ps") "ch") ("ancoms" "ps" "ph" "ch" ("anabsi5" "ps" "ph" "ch" "anabsi8_1"))) ;; Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.) (theorem "anabss7" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anabss7_1" ("wi" ("wa" "ps" ("wa" "ph" "ps")) "ch"))) (for) ("wi" ("wa" "ph" "ps") "ch") ("anabss4" "ph" "ps" "ch" ("anassrs" "ps" "ph" "ps" "ch" "anabss7_1"))) ;; Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (theorem "anabsan2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anabsan2_1" ("wi" ("wa" "ph" ("wa" "ps" "ps")) "ch"))) (for) ("wi" ("wa" "ph" "ps") "ch") ("anabss7" "ph" "ps" "ch" ("an12s" "ph" "ps" "ps" "ch" "anabsan2_1"))) ;; Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.) (theorem "anabss3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("anabss3_1" ("wi" ("wa" ("wa" "ph" "ps") "ps") "ch"))) (for) ("wi" ("wa" "ph" "ps") "ch") ("anabsan2" "ph" "ps" "ch" ("anasss" "ph" "ps" "ps" "ch" "anabss3_1"))) ;; Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.) (theorem "an4" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) ("wa" ("wa" "ph" "ch") ("wa" "ps" "th"))) ("_3bitr4i" ("wa" "ph" ("wa" "ps" ("wa" "ch" "th"))) ("wa" "ph" ("wa" "ch" ("wa" "ps" "th"))) ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) ("wa" ("wa" "ph" "ch") ("wa" "ps" "th")) ("anbi2i" ("wa" "ps" ("wa" "ch" "th")) ("wa" "ch" ("wa" "ps" "th")) "ph" ("an12" "ps" "ch" "th")) ("anass" "ph" "ps" ("wa" "ch" "th")) ("anass" "ph" "ch" ("wa" "ps" "th")))) ;; Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.) (theorem "an42" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) ("wa" ("wa" "ph" "ch") ("wa" "th" "ps"))) ("bitri" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) ("wa" ("wa" "ph" "ch") ("wa" "ps" "th")) ("wa" ("wa" "ph" "ch") ("wa" "th" "ps")) ("an4" "ph" "ps" "ch" "th") ("anbi2i" ("wa" "ps" "th") ("wa" "th" "ps") ("wa" "ph" "ch") ("ancom" "ps" "th")))) ;; Rearrangement of 4 conjuncts. (Contributed by Rodolfo Medina, 24-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (theorem "an43" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) ("wa" ("wa" "ph" "th") ("wa" "ps" "ch"))) ("bicomi" ("wa" ("wa" "ph" "th") ("wa" "ps" "ch")) ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) ("an42" "ph" "th" "ps" "ch"))) ;; A rearrangement of conjuncts. (Contributed by Rodolfo Medina, 25-Sep-2010.) (theorem "an3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) ("wa" "ph" "th")) ("simplbi" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) ("wa" "ph" "th") ("wa" "ps" "ch") ("an43" "ph" "ps" "ch" "th"))) ;; Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.) (theorem "an4s" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("an4s_1" ("wi" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) "ta"))) (for) ("wi" ("wa" ("wa" "ph" "ch") ("wa" "ps" "th")) "ta") ("sylbi" ("wa" ("wa" "ph" "ch") ("wa" "ps" "th")) ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) "ta" ("an4" "ph" "ch" "ps" "th") "an4s_1")) ;; Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.) (theorem "an42s" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("an41r3s_1" ("wi" ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) "ta"))) (for) ("wi" ("wa" ("wa" "ph" "ch") ("wa" "th" "ps")) "ta") ("ancom2s" ("wa" "ph" "ch") "ps" "th" "ta" ("an4s" "ph" "ps" "ch" "th" "ta" "an41r3s_1"))) ;; Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.) (theorem "anandi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wa" "ps" "ch")) ("wa" ("wa" "ph" "ps") ("wa" "ph" "ch"))) ("bitr3i" ("wa" "ph" ("wa" "ps" "ch")) ("wa" ("wa" "ph" "ph") ("wa" "ps" "ch")) ("wa" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("anbi1i" ("wa" "ph" "ph") "ph" ("wa" "ps" "ch") ("anidm" "ph")) ("an4" "ph" "ph" "ps" "ch"))) ;; Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.) (theorem "anandir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" ("wa" "ph" "ps") "ch") ("wa" ("wa" "ph" "ch") ("wa" "ps" "ch"))) ("bitr3i" ("wa" ("wa" "ph" "ps") "ch") ("wa" ("wa" "ph" "ps") ("wa" "ch" "ch")) ("wa" ("wa" "ph" "ch") ("wa" "ps" "ch")) ("anbi2i" ("wa" "ch" "ch") "ch" ("wa" "ph" "ps") ("anidm" "ch")) ("an4" "ph" "ps" "ch" "ch"))) ;; Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.) (theorem "anandis" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("ta" ( "wff"))) (for ("anandis_1" ("wi" ("wa" ("wa" "ph" "ps") ("wa" "ph" "ch")) "ta"))) (for) ("wi" ("wa" "ph" ("wa" "ps" "ch")) "ta") ("anabsan" "ph" ("wa" "ps" "ch") "ta" ("an4s" "ph" "ps" "ph" "ch" "ta" "anandis_1"))) ;; Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.) (theorem "anandirs" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("ta" ( "wff"))) (for ("anandirs_1" ("wi" ("wa" ("wa" "ph" "ch") ("wa" "ps" "ch")) "ta"))) (for) ("wi" ("wa" ("wa" "ph" "ps") "ch") "ta") ("anabsan2" ("wa" "ph" "ps") "ch" "ta" ("an4s" "ph" "ch" "ps" "ch" "ta" "anandirs_1"))) ;; ~ syl2an with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.) (theorem "syl2an2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl2an2_1" ("wi" "ph" "ps")) ("syl2an2_2" ("wi" ("wa" "ch" "ph") "th")) ("syl2an2_3" ("wi" ("wa" "ps" "th") "ta"))) (for) ("wi" ("wa" "ch" "ph") "ta") ("anabss7" "ch" "ph" "ta" ("syl2an" "ph" "ps" "th" "ta" ("wa" "ch" "ph") "syl2an2_1" "syl2an2_2" "syl2an2_3"))) ;; ~ syl2anr with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.) (Proof shortened by Wolf Lammen, 28-Mar-2022.) (theorem "syl2an2r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl2an2r_1" ("wi" "ph" "ps")) ("syl2an2r_2" ("wi" ("wa" "ph" "ch") "th")) ("syl2an2r_3" ("wi" ("wa" "ps" "th") "ta"))) (for) ("wi" ("wa" "ph" "ch") "ta") ("syldan" "ph" "ch" "th" "ta" "syl2an2r_2" ("sylan" "ph" "ps" "th" "ta" "syl2an2r_1" "syl2an2r_3"))) ;; Obsolete proof of ~ syl2an2r as of 28-Mar-2022. (Contributed by Alan Sare, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "syl2an2rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl2an2r_1" ("wi" "ph" "ps")) ("syl2an2r_2" ("wi" ("wa" "ph" "ch") "th")) ("syl2an2r_3" ("wi" ("wa" "ps" "th") "ta"))) (for) ("wi" ("wa" "ph" "ch") "ta") ("anabss5" "ph" "ch" "ta" ("syl2an" "ph" "ps" "th" "ta" ("wa" "ph" "ch") "syl2an2r_1" "syl2an2r_2" "syl2an2r_3"))) ;; Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.) (theorem "impbida" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("impbida_1" ("wi" ("wa" "ph" "ps") "ch")) ("impbida_2" ("wi" ("wa" "ph" "ch") "ps"))) (for) ("wi" "ph" ("wb" "ps" "ch")) ("impbid" "ph" "ps" "ch" ("ex" "ph" "ps" "ch" "impbida_1") ("ex" "ph" "ch" "ps" "impbida_2"))) ;; Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) (theorem "pm3_48" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wi" "ph" "ps") ("wi" "ch" "th")) ("wi" ("wo" "ph" "ch") ("wo" "ps" "th"))) ("jaao" ("wi" "ph" "ps") "ph" ("wo" "ps" "th") ("wi" "ch" "th") "ch" ("imim2i" "ps" ("wo" "ps" "th") "ph" ("orc" "ps" "th")) ("imim2i" "th" ("wo" "ps" "th") "ch" ("olc" "th" "ps")))) ;; Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (theorem "pm3_45" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wa" "ph" "ch") ("wa" "ps" "ch"))) ("anim1d" ("wi" "ph" "ps") "ph" "ps" "ch" ("id" ("wi" "ph" "ps")))) ;; Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.) (theorem "im2anan9" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("im2an9_1" ("wi" "ph" ("wi" "ps" "ch"))) ("im2an9_2" ("wi" "th" ("wi" "ta" "et")))) (for) ("wi" ("wa" "ph" "th") ("wi" ("wa" "ps" "ta") ("wa" "ch" "et"))) ("anim12d" ("wa" "ph" "th") "ps" "ch" "ta" "et" ("adantr" "ph" ("wi" "ps" "ch") "th" "im2an9_1") ("adantl" "th" ("wi" "ta" "et") "ph" "im2an9_2"))) ;; Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.) (theorem "im2anan9r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("im2an9_1" ("wi" "ph" ("wi" "ps" "ch"))) ("im2an9_2" ("wi" "th" ("wi" "ta" "et")))) (for) ("wi" ("wa" "th" "ph") ("wi" ("wa" "ps" "ta") ("wa" "ch" "et"))) ("ancoms" "ph" "th" ("wi" ("wa" "ps" "ta") ("wa" "ch" "et")) ("im2anan9" "ph" "ps" "ch" "th" "ta" "et" "im2an9_1" "im2an9_2"))) ;; Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.) (theorem "anim12dan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("anim12dan_1" ("wi" ("wa" "ph" "ps") "ch")) ("anim12dan_2" ("wi" ("wa" "ph" "th") "ta"))) (for) ("wi" ("wa" "ph" ("wa" "ps" "th")) ("wa" "ch" "ta")) ("imp" "ph" ("wa" "ps" "th") ("wa" "ch" "ta") ("anim12d" "ph" "ps" "ch" "th" "ta" ("ex" "ph" "ps" "ch" "anim12dan_1") ("ex" "ph" "th" "ta" "anim12dan_2")))) ;; Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.) (theorem "orim12d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("orim12d_1" ("wi" "ph" ("wi" "ps" "ch"))) ("orim12d_2" ("wi" "ph" ("wi" "th" "ta")))) (for) ("wi" "ph" ("wi" ("wo" "ps" "th") ("wo" "ch" "ta"))) ("syl2anc" "ph" ("wi" "ps" "ch") ("wi" "th" "ta") ("wi" ("wo" "ps" "th") ("wo" "ch" "ta")) "orim12d_1" "orim12d_2" ("pm3_48" "ps" "ch" "th" "ta"))) ;; Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.) (theorem "orim1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("orim1d_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" ("wo" "ps" "th") ("wo" "ch" "th"))) ("orim12d" "ph" "ps" "ch" "th" "th" "orim1d_1" ("idd" "ph" "th"))) ;; Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.) (theorem "orim2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("orim1d_1" ("wi" "ph" ("wi" "ps" "ch")))) (for) ("wi" "ph" ("wi" ("wo" "th" "ps") ("wo" "th" "ch"))) ("orim12d" "ph" "th" "th" "ps" "ch" ("idd" "ph" "th") "orim1d_1")) ;; Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.) (theorem "orim2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ps" "ch") ("wi" ("wo" "ph" "ps") ("wo" "ph" "ch"))) ("orim2d" ("wi" "ps" "ch") "ps" "ch" "ph" ("id" ("wi" "ps" "ch")))) ;; Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) (theorem "pm2_38" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ps" "ch") ("wi" ("wo" "ps" "ph") ("wo" "ch" "ph"))) ("orim1d" ("wi" "ps" "ch") "ps" "ch" "ph" ("id" ("wi" "ps" "ch")))) ;; Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) (theorem "pm2_36" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ps" "ch") ("wi" ("wo" "ph" "ps") ("wo" "ch" "ph"))) ("syl5" ("wo" "ph" "ps") ("wo" "ps" "ph") ("wi" "ps" "ch") ("wo" "ch" "ph") ("pm1_4" "ph" "ps") ("pm2_38" "ph" "ps" "ch"))) ;; Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) (theorem "pm2_37" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ps" "ch") ("wi" ("wo" "ps" "ph") ("wo" "ph" "ch"))) ("syl6" ("wi" "ps" "ch") ("wo" "ps" "ph") ("wo" "ch" "ph") ("wo" "ph" "ch") ("pm2_38" "ph" "ps" "ch") ("pm1_4" "ch" "ph"))) ;; Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_73" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wo" ("wo" "ph" "ps") "ch") ("wo" "ps" "ch"))) ("orim1d" ("wi" "ph" "ps") ("wo" "ph" "ps") "ps" "ch" ("pm2_621" "ph" "ps"))) ;; Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "pm2_74" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ps" "ph") ("wi" ("wo" ("wo" "ph" "ps") "ch") ("wo" "ph" "ch"))) ("orim1d" ("wi" "ps" "ph") ("wo" "ph" "ps") "ph" "ch" ("ja" "ps" "ph" ("wi" ("wo" "ph" "ps") "ph") ("orel2" "ps" "ph") ("ax_1" "ph" ("wo" "ph" "ps"))))) ;; Disjunction distributes over implication. (Contributed by Wolf Lammen, 5-Jan-2013.) (theorem "orimdi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" "ph" ("wi" "ps" "ch")) ("wi" ("wo" "ph" "ps") ("wo" "ph" "ch"))) ("_3bitr4i" ("wi" ("wn" "ph") ("wi" "ps" "ch")) ("wi" ("wi" ("wn" "ph") "ps") ("wi" ("wn" "ph") "ch")) ("wo" "ph" ("wi" "ps" "ch")) ("wi" ("wo" "ph" "ps") ("wo" "ph" "ch")) ("imdi" ("wn" "ph") "ps" "ch") ("df_or" "ph" ("wi" "ps" "ch")) ("imbi12i" ("wo" "ph" "ps") ("wi" ("wn" "ph") "ps") ("wo" "ph" "ch") ("wi" ("wn" "ph") "ch") ("df_or" "ph" "ps") ("df_or" "ph" "ch")))) ;; Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_76" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wo" "ph" ("wi" "ps" "ch")) ("wi" ("wo" "ph" "ps") ("wo" "ph" "ch"))) ("biimpi" ("wo" "ph" ("wi" "ps" "ch")) ("wi" ("wo" "ph" "ps") ("wo" "ph" "ch")) ("orimdi" "ph" "ps" "ch"))) ;; Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.) (theorem "pm2_75" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wo" "ph" "ps") ("wi" ("wo" "ph" ("wi" "ps" "ch")) ("wo" "ph" "ch"))) ("com12" ("wo" "ph" ("wi" "ps" "ch")) ("wo" "ph" "ps") ("wo" "ph" "ch") ("pm2_76" "ph" "ps" "ch"))) ;; Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) (theorem "pm2_8" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wo" "ph" "ps") ("wi" ("wo" ("wn" "ps") "ch") ("wo" "ph" "ch"))) ("orim1d" ("wo" "ph" "ps") ("wn" "ps") "ph" "ch" ("con1d" ("wo" "ph" "ps") "ph" "ps" ("pm2_53" "ph" "ps")))) ;; Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_81" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wi" "ps" ("wi" "ch" "th")) ("wi" ("wo" "ph" "ps") ("wi" ("wo" "ph" "ch") ("wo" "ph" "th")))) ("syl6" ("wi" "ps" ("wi" "ch" "th")) ("wo" "ph" "ps") ("wo" "ph" ("wi" "ch" "th")) ("wi" ("wo" "ph" "ch") ("wo" "ph" "th")) ("orim2" "ph" "ps" ("wi" "ch" "th")) ("pm2_76" "ph" "ch" "th"))) ;; Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (theorem "pm2_82" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wo" ("wo" "ph" "ps") "ch") ("wi" ("wo" ("wo" "ph" ("wn" "ch")) "th") ("wo" ("wo" "ph" "ps") "th"))) ("orim1d" ("wo" ("wo" "ph" "ps") "ch") ("wo" "ph" ("wn" "ch")) ("wo" "ph" "ps") "th" ("jao1i" ("wo" "ph" "ps") "ch" ("wo" "ph" ("wn" "ch")) ("orim2d" "ch" ("wn" "ch") "ps" "ph" ("pm2_24" "ch" "ps"))))) ;; Theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) (theorem "pm2_85" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" ("wo" "ph" "ps") ("wo" "ph" "ch")) ("wo" "ph" ("wi" "ps" "ch"))) ("biimpri" ("wo" "ph" ("wi" "ps" "ch")) ("wi" ("wo" "ph" "ps") ("wo" "ph" "ch")) ("orimdi" "ph" "ps" "ch"))) ;; Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.) (theorem "pm3_2ni" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm3_2ni_1" ("wn" "ph")) ("pm3_2ni_2" ("wn" "ps"))) (for) ("wn" ("wo" "ph" "ps")) ("mto" ("wo" "ph" "ps") "ph" "pm3_2ni_1" ("jaoi" "ph" "ph" "ps" ("id" "ph") ("pm2_21i" "ps" "ph" "pm3_2ni_2")))) ;; Absorption of redundant internal disjunct. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 28-Feb-2014.) (theorem "orabs" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" "ph" ("wa" ("wo" "ph" "ps") "ph")) ("pm4_71ri" "ph" ("wo" "ph" "ps") ("orc" "ph" "ps"))) ;; Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.) (theorem "oranabs" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" ("wo" "ph" ("wn" "ps")) "ps") ("wa" "ph" "ps")) ("pm5_32ri" "ps" ("wo" "ph" ("wn" "ps")) "ph" ("syl6rbb" "ps" "ph" ("wo" ("wn" "ps") "ph") ("wo" "ph" ("wn" "ps")) ("biortn" "ps" "ph") ("orcom" ("wn" "ps") "ph")))) ;; Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.) (theorem "pm5_1" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") ("wb" "ph" "ps")) ("biimpa" "ph" "ps" ("wb" "ph" "ps") ("pm5_501" "ph" "ps"))) ;; Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.) (theorem "pm5_21" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" ("wn" "ph") ("wn" "ps")) ("wb" "ph" "ps")) ("imp" ("wn" "ph") ("wn" "ps") ("wb" "ph" "ps") ("pm5_21im" "ph" "ps"))) ;; If neither of two propositions is true, then these propositions are equivalent. (Contributed by BJ, 26-Apr-2019.) (theorem "norbi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wo" "ph" "ps")) ("wb" "ph" "ps")) ("pm5_21ni" "ph" ("wo" "ph" "ps") "ps" ("orc" "ph" "ps") ("olc" "ps" "ph"))) ;; If two propositions are not equivalent, then at least one is true. (Contributed by BJ, 19-Apr-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.) (theorem "nbior" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wn" ("wb" "ph" "ps")) ("wo" "ph" "ps")) ("con1i" ("wo" "ph" "ps") ("wb" "ph" "ps") ("norbi" "ph" "ps"))) ;; Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (theorem "pm3_43" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("wi" "ph" ("wa" "ps" "ch"))) ("imp" ("wi" "ph" "ps") ("wi" "ph" "ch") ("wi" "ph" ("wa" "ps" "ch")) ("pm3_43i" "ph" "ps" "ch"))) ;; Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.) (theorem "jcab" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" "ph" ("wa" "ps" "ch")) ("wa" ("wi" "ph" "ps") ("wi" "ph" "ch"))) ("impbii" ("wi" "ph" ("wa" "ps" "ch")) ("wa" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("jca" ("wi" "ph" ("wa" "ps" "ch")) ("wi" "ph" "ps") ("wi" "ph" "ch") ("imim2i" ("wa" "ps" "ch") "ps" "ph" ("simpl" "ps" "ch")) ("imim2i" ("wa" "ps" "ch") "ch" "ph" ("simpr" "ps" "ch"))) ("pm3_43" "ph" "ps" "ch"))) ;; Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 28-Nov-2013.) (theorem "ordi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" "ph" ("wa" "ps" "ch")) ("wa" ("wo" "ph" "ps") ("wo" "ph" "ch"))) ("_3bitr4i" ("wi" ("wn" "ph") ("wa" "ps" "ch")) ("wa" ("wi" ("wn" "ph") "ps") ("wi" ("wn" "ph") "ch")) ("wo" "ph" ("wa" "ps" "ch")) ("wa" ("wo" "ph" "ps") ("wo" "ph" "ch")) ("jcab" ("wn" "ph") "ps" "ch") ("df_or" "ph" ("wa" "ps" "ch")) ("anbi12i" ("wo" "ph" "ps") ("wi" ("wn" "ph") "ps") ("wo" "ph" "ch") ("wi" ("wn" "ph") "ch") ("df_or" "ph" "ps") ("df_or" "ph" "ch")))) ;; Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) (theorem "ordir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" ("wa" "ph" "ps") "ch") ("wa" ("wo" "ph" "ch") ("wo" "ps" "ch"))) ("_3bitr4i" ("wo" "ch" ("wa" "ph" "ps")) ("wa" ("wo" "ch" "ph") ("wo" "ch" "ps")) ("wo" ("wa" "ph" "ps") "ch") ("wa" ("wo" "ph" "ch") ("wo" "ps" "ch")) ("ordi" "ch" "ph" "ps") ("orcom" ("wa" "ph" "ps") "ch") ("anbi12i" ("wo" "ph" "ch") ("wo" "ch" "ph") ("wo" "ps" "ch") ("wo" "ch" "ps") ("orcom" "ph" "ch") ("orcom" "ps" "ch")))) ;; Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_76" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("wi" "ph" ("wa" "ps" "ch"))) ("bicomi" ("wi" "ph" ("wa" "ps" "ch")) ("wa" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("jcab" "ph" "ps" "ch"))) ;; Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) (theorem "andi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wo" "ps" "ch")) ("wo" ("wa" "ph" "ps") ("wa" "ph" "ch"))) ("impbii" ("wa" "ph" ("wo" "ps" "ch")) ("wo" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("jaodan" "ph" "ps" ("wo" ("wa" "ph" "ps") ("wa" "ph" "ch")) "ch" ("orc" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("olc" ("wa" "ph" "ch") ("wa" "ph" "ps"))) ("jaoi" ("wa" "ph" "ps") ("wa" "ph" ("wo" "ps" "ch")) ("wa" "ph" "ch") ("anim2i" "ps" ("wo" "ps" "ch") "ph" ("orc" "ps" "ch")) ("anim2i" "ch" ("wo" "ps" "ch") "ph" ("olc" "ch" "ps"))))) ;; Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) (theorem "andir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" ("wo" "ph" "ps") "ch") ("wo" ("wa" "ph" "ch") ("wa" "ps" "ch"))) ("_3bitr4i" ("wa" "ch" ("wo" "ph" "ps")) ("wo" ("wa" "ch" "ph") ("wa" "ch" "ps")) ("wa" ("wo" "ph" "ps") "ch") ("wo" ("wa" "ph" "ch") ("wa" "ps" "ch")) ("andi" "ch" "ph" "ps") ("ancom" ("wo" "ph" "ps") "ch") ("orbi12i" ("wa" "ph" "ch") ("wa" "ch" "ph") ("wa" "ps" "ch") ("wa" "ch" "ps") ("ancom" "ph" "ch") ("ancom" "ps" "ch")))) ;; Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) (theorem "orddi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wo" ("wa" "ph" "ps") ("wa" "ch" "th")) ("wa" ("wa" ("wo" "ph" "ch") ("wo" "ph" "th")) ("wa" ("wo" "ps" "ch") ("wo" "ps" "th")))) ("bitri" ("wo" ("wa" "ph" "ps") ("wa" "ch" "th")) ("wa" ("wo" "ph" ("wa" "ch" "th")) ("wo" "ps" ("wa" "ch" "th"))) ("wa" ("wa" ("wo" "ph" "ch") ("wo" "ph" "th")) ("wa" ("wo" "ps" "ch") ("wo" "ps" "th"))) ("ordir" "ph" "ps" ("wa" "ch" "th")) ("anbi12i" ("wo" "ph" ("wa" "ch" "th")) ("wa" ("wo" "ph" "ch") ("wo" "ph" "th")) ("wo" "ps" ("wa" "ch" "th")) ("wa" ("wo" "ps" "ch") ("wo" "ps" "th")) ("ordi" "ph" "ch" "th") ("ordi" "ps" "ch" "th")))) ;; Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) (theorem "anddi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wa" ("wo" "ph" "ps") ("wo" "ch" "th")) ("wo" ("wo" ("wa" "ph" "ch") ("wa" "ph" "th")) ("wo" ("wa" "ps" "ch") ("wa" "ps" "th")))) ("bitri" ("wa" ("wo" "ph" "ps") ("wo" "ch" "th")) ("wo" ("wa" "ph" ("wo" "ch" "th")) ("wa" "ps" ("wo" "ch" "th"))) ("wo" ("wo" ("wa" "ph" "ch") ("wa" "ph" "th")) ("wo" ("wa" "ps" "ch") ("wa" "ps" "th"))) ("andir" "ph" "ps" ("wo" "ch" "th")) ("orbi12i" ("wa" "ph" ("wo" "ch" "th")) ("wo" ("wa" "ph" "ch") ("wa" "ph" "th")) ("wa" "ps" ("wo" "ch" "th")) ("wo" ("wa" "ps" "ch") ("wa" "ps" "th")) ("andi" "ph" "ch" "th") ("andi" "ps" "ch" "th")))) ;; Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_39" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wb" "ph" "ch") ("wb" "ps" "th")) ("wb" ("wo" "ph" "ps") ("wo" "ch" "th"))) ("orbi12d" ("wa" ("wb" "ph" "ch") ("wb" "ps" "th")) "ph" "ch" "ps" "th" ("simpl" ("wb" "ph" "ch") ("wb" "ps" "th")) ("simpr" ("wb" "ph" "ch") ("wb" "ps" "th")))) ;; Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_38" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wb" "ph" "ch") ("wb" "ps" "th")) ("wb" ("wa" "ph" "ps") ("wa" "ch" "th"))) ("anbi12d" ("wa" ("wb" "ph" "ch") ("wb" "ps" "th")) "ph" "ch" "ps" "th" ("simpl" ("wb" "ph" "ch") ("wb" "ps" "th")) ("simpr" ("wb" "ph" "ch") ("wb" "ps" "th")))) ;; Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.) (theorem "bi2anan9" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("bi2an9_1" ("wi" "ph" ("wb" "ps" "ch"))) ("bi2an9_2" ("wi" "th" ("wb" "ta" "et")))) (for) ("wi" ("wa" "ph" "th") ("wb" ("wa" "ps" "ta") ("wa" "ch" "et"))) ("sylan9bb" "ph" ("wa" "ps" "ta") ("wa" "ch" "ta") "th" ("wa" "ch" "et") ("anbi1d" "ph" "ps" "ch" "ta" "bi2an9_1") ("anbi2d" "th" "ta" "et" "ch" "bi2an9_2"))) ;; Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.) (theorem "bi2anan9r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("bi2an9_1" ("wi" "ph" ("wb" "ps" "ch"))) ("bi2an9_2" ("wi" "th" ("wb" "ta" "et")))) (for) ("wi" ("wa" "th" "ph") ("wb" ("wa" "ps" "ta") ("wa" "ch" "et"))) ("ancoms" "ph" "th" ("wb" ("wa" "ps" "ta") ("wa" "ch" "et")) ("bi2anan9" "ph" "ps" "ch" "th" "ta" "et" "bi2an9_1" "bi2an9_2"))) ;; Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.) (theorem "bi2bian9" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("bi2an9_1" ("wi" "ph" ("wb" "ps" "ch"))) ("bi2an9_2" ("wi" "th" ("wb" "ta" "et")))) (for) ("wi" ("wa" "ph" "th") ("wb" ("wb" "ps" "ta") ("wb" "ch" "et"))) ("bibi12d" ("wa" "ph" "th") "ps" "ch" "ta" "et" ("adantr" "ph" ("wb" "ps" "ch") "th" "bi2an9_1") ("adantl" "th" ("wb" "ta" "et") "ph" "bi2an9_2"))) ;; Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.) (theorem "pm4_72" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" "ps") ("wb" "ps" ("wo" "ph" "ps"))) ("impbii" ("wi" "ph" "ps") ("wb" "ps" ("wo" "ph" "ps")) ("impbid2" ("wi" "ph" "ps") "ps" ("wo" "ph" "ps") ("olc" "ps" "ph") ("pm2_621" "ph" "ps")) ("syl5" "ph" ("wo" "ph" "ps") ("wb" "ps" ("wo" "ph" "ps")) "ps" ("orc" "ph" "ps") ("biimpr" "ps" ("wo" "ph" "ps"))))) ;; Simplify an implication between implications. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) (theorem "imimorb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" ("wi" "ps" "ch") ("wi" "ph" "ch")) ("wi" "ph" ("wo" "ps" "ch"))) ("bitr4i" ("wi" ("wi" "ps" "ch") ("wi" "ph" "ch")) ("wi" "ph" ("wi" ("wi" "ps" "ch") "ch")) ("wi" "ph" ("wo" "ps" "ch")) ("bi2_04" ("wi" "ps" "ch") "ph" "ch") ("imbi2i" ("wo" "ps" "ch") ("wi" ("wi" "ps" "ch") "ch") "ph" ("dfor2" "ps" "ch")))) ;; Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_33" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wi" "ps" "ch")) ("wa" "ph" ("wi" ("wa" "ph" "ps") "ch"))) ("pm5_32i" "ph" ("wi" "ps" "ch") ("wi" ("wa" "ph" "ps") "ch") ("imbi1d" "ph" "ps" ("wa" "ph" "ps") "ch" ("ibar" "ph" "ps")))) ;; Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_36" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wb" "ph" "ps")) ("wa" "ps" ("wb" "ph" "ps"))) ("pm5_32ri" ("wb" "ph" "ps") "ph" "ps" ("id" ("wb" "ph" "ps")))) ;; Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.) (theorem "bianabs" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("bianabs_1" ("wi" "ph" ("wb" "ps" ("wa" "ph" "ch"))))) (for) ("wi" "ph" ("wb" "ps" "ch")) ("bitr4d" "ph" "ps" ("wa" "ph" "ch") "ch" "bianabs_1" ("ibar" "ph" "ch"))) ;; Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) (theorem "oibabs" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" ("wo" "ph" "ps") ("wb" "ph" "ps")) ("wb" "ph" "ps")) ("impbii" ("wi" ("wo" "ph" "ps") ("wb" "ph" "ps")) ("wb" "ph" "ps") ("ja" ("wo" "ph" "ps") ("wb" "ph" "ps") ("wb" "ph" "ps") ("norbi" "ph" "ps") ("id" ("wb" "ph" "ps"))) ("ax_1" ("wb" "ph" "ps") ("wo" "ph" "ps")))) ;; Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) (theorem "pm3_24" (for ("ph" ( "wff"))) (for) (for) ("wn" ("wa" "ph" ("wn" "ph"))) ("mpbi" ("wi" "ph" "ph") ("wn" ("wa" "ph" ("wn" "ph"))) ("id" "ph") ("iman" "ph" "ph"))) ;; Theorem *2.26 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) (theorem "pm2_26" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wo" ("wn" "ph") ("wi" ("wi" "ph" "ps") "ps")) ("imori" "ph" ("wi" ("wi" "ph" "ps") "ps") ("pm2_27" "ph" "ps"))) ;; Theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_11" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wo" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ps")) ("orri" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ps") ("pm2_5" "ph" "ps"))) ;; Theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_12" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wo" ("wi" "ph" "ps") ("wi" "ph" ("wn" "ps"))) ("orri" ("wi" "ph" "ps") ("wi" "ph" ("wn" "ps")) ("pm2_51" "ph" "ps"))) ;; Theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_14" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wo" ("wi" "ph" "ps") ("wi" "ps" "ch")) ("orri" ("wi" "ph" "ps") ("wi" "ps" "ch") ("pm2_21d" ("wn" ("wi" "ph" "ps")) "ps" "ch" ("con3i" "ps" ("wi" "ph" "ps") ("ax_1" "ps" "ph"))))) ;; Theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) (theorem "pm5_13" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wo" ("wi" "ph" "ps") ("wi" "ps" "ph")) ("pm5_14" "ph" "ps" "ph")) ;; Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.) (theorem "pm5_17" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" ("wo" "ph" "ps") ("wn" ("wa" "ph" "ps"))) ("wb" "ph" ("wn" "ps"))) ("_3bitrri" ("wb" "ph" ("wn" "ps")) ("wb" ("wn" "ps") "ph") ("wa" ("wi" ("wn" "ps") "ph") ("wi" "ph" ("wn" "ps"))) ("wa" ("wo" "ph" "ps") ("wn" ("wa" "ph" "ps"))) ("bicom" "ph" ("wn" "ps")) ("dfbi2" ("wn" "ps") "ph") ("anbi12i" ("wi" ("wn" "ps") "ph") ("wo" "ph" "ps") ("wi" "ph" ("wn" "ps")) ("wn" ("wa" "ph" "ps")) ("bitr2i" ("wo" "ph" "ps") ("wo" "ps" "ph") ("wi" ("wn" "ps") "ph") ("orcom" "ph" "ps") ("df_or" "ps" "ph")) ("imnan" "ph" "ps")))) ;; Theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 15-Oct-2013.) (theorem "pm5_15" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wo" ("wb" "ph" "ps") ("wb" "ph" ("wn" "ps"))) ("orri" ("wb" "ph" "ps") ("wb" "ph" ("wn" "ps")) ("biimpi" ("wn" ("wb" "ph" "ps")) ("wb" "ph" ("wn" "ps")) ("xor3" "ph" "ps")))) ;; Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 17-Oct-2013.) (theorem "pm5_16" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wn" ("wa" ("wb" "ph" "ps") ("wb" "ph" ("wn" "ps")))) ("mpbi" ("wi" ("wb" "ph" "ps") ("wn" ("wb" "ph" ("wn" "ps")))) ("wn" ("wa" ("wb" "ph" "ps") ("wb" "ph" ("wn" "ps")))) ("biimpi" ("wb" "ph" "ps") ("wn" ("wb" "ph" ("wn" "ps"))) ("pm5_18" "ph" "ps")) ("imnan" ("wb" "ph" "ps") ("wb" "ph" ("wn" "ps"))))) ;; Two ways to express "exclusive or." Theorem *5.22 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 22-Jan-2013.) (theorem "xor" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wb" "ph" "ps")) ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph")))) ("con1bii" ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph"))) ("wb" "ph" "ps") ("_3bitr4ri" ("wa" ("wi" "ph" "ps") ("wi" "ps" "ph")) ("wa" ("wn" ("wa" "ph" ("wn" "ps"))) ("wn" ("wa" "ps" ("wn" "ph")))) ("wb" "ph" "ps") ("wn" ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph")))) ("anbi12i" ("wi" "ph" "ps") ("wn" ("wa" "ph" ("wn" "ps"))) ("wi" "ps" "ph") ("wn" ("wa" "ps" ("wn" "ph"))) ("iman" "ph" "ps") ("iman" "ps" "ph")) ("dfbi2" "ph" "ps") ("ioran" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph")))))) ;; Two ways to express "exclusive or." (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Jan-2013.) (theorem "nbi2" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wb" "ph" "ps")) ("wa" ("wo" "ph" "ps") ("wn" ("wa" "ph" "ps")))) ("bitr4i" ("wn" ("wb" "ph" "ps")) ("wb" "ph" ("wn" "ps")) ("wa" ("wo" "ph" "ps") ("wn" ("wa" "ph" "ps"))) ("xor3" "ph" "ps") ("pm5_17" "ph" "ps"))) ;; Conjunction distributes over exclusive-or, using ` -. ( ph <-> ps ) ` to express exclusive-or. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. This is not necessarily true in intuitionistic logic, though ~ anxordi does hold in it. (Contributed by NM, 3-Oct-2008.) (theorem "xordi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wn" ("wb" "ps" "ch"))) ("wn" ("wb" ("wa" "ph" "ps") ("wa" "ph" "ch")))) ("xchbinx" ("wa" "ph" ("wn" ("wb" "ps" "ch"))) ("wi" "ph" ("wb" "ps" "ch")) ("wb" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("annim" "ph" ("wb" "ps" "ch")) ("pm5_32" "ph" "ps" "ch"))) ;; A wff disjoined with truth is true. (Contributed by NM, 23-May-1999.) (theorem "biort" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wb" "ph" ("wo" "ph" "ps"))) ("impbid2" "ph" "ph" ("wo" "ph" "ps") ("orc" "ph" "ps") ("ax_1" "ph" ("wo" "ph" "ps")))) ;; Theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 20-Jan-2013.) (theorem "pm5_55" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wo" ("wb" ("wo" "ph" "ps") "ph") ("wb" ("wo" "ph" "ps") "ps")) ("orri" ("wb" ("wo" "ph" "ps") "ph") ("wb" ("wo" "ph" "ps") "ps") ("con1i" ("wb" ("wo" "ph" "ps") "ps") ("wb" ("wo" "ph" "ps") "ph") ("nsyl4" "ph" ("wb" ("wo" "ph" "ps") "ph") ("wb" ("wo" "ph" "ps") "ps") ("bicomd" "ph" "ph" ("wo" "ph" "ps") ("biort" "ph" "ps")) ("bicomd" ("wn" "ph") "ps" ("wo" "ph" "ps") ("biorf" "ph" "ps")))))) ;; Selecting one statement from a disjunction if one of the disjuncted statements is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.) (Proof shortened by Wolf Lammen, 19-Jan-2020.) (theorem "ornld" (for ("ph" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" "ph" ("wi" ("wa" ("wi" "ph" ("wo" "th" "ta")) ("wn" "th")) "ta")) ("expimpd" "ph" ("wi" "ph" ("wo" "th" "ta")) ("wn" "th") "ta" ("ord" ("wa" "ph" ("wi" "ph" ("wo" "th" "ta"))) "th" "ta" ("pm3_35" "ph" ("wo" "th" "ta"))))) ;; A conjunction with a negated conjunction. (Contributed by AV, 8-Mar-2022.) (Proof shortened by Wolf Lammen, 1-Apr-2022.) (theorem "annotanannot" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wn" ("wa" "ph" "ps"))) ("wa" "ph" ("wn" "ps"))) ("pm5_32i" "ph" ("wn" ("wa" "ph" "ps")) ("wn" "ps") ("notbid" "ph" ("wa" "ph" "ps") "ps" ("bicomd" "ph" "ps" ("wa" "ph" "ps") ("ibar" "ph" "ps"))))) ;; Obsolete proof of ~ annotanannot as of 1-Apr-2022. (Contributed by AV, 8-Mar-2022.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "annotanannotOLD" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wn" ("wa" "ph" "ps"))) ("wa" "ph" ("wn" "ps"))) ("_3bitri" ("wa" "ph" ("wn" ("wa" "ph" "ps"))) ("wa" "ph" ("wo" ("wn" "ph") ("wn" "ps"))) ("wo" ("wa" "ph" ("wn" "ph")) ("wa" "ph" ("wn" "ps"))) ("wa" "ph" ("wn" "ps")) ("anbi2i" ("wn" ("wa" "ph" "ps")) ("wo" ("wn" "ph") ("wn" "ps")) "ph" ("ianor" "ph" "ps")) ("andi" "ph" ("wn" "ph") ("wn" "ps")) ("impbii" ("wo" ("wa" "ph" ("wn" "ph")) ("wa" "ph" ("wn" "ps"))) ("wa" "ph" ("wn" "ps")) ("jaoi" ("wa" "ph" ("wn" "ph")) ("wa" "ph" ("wn" "ps")) ("wa" "ph" ("wn" "ps")) ("pm2_21i" ("wa" "ph" ("wn" "ph")) ("wa" "ph" ("wn" "ps")) ("pm3_24" "ph")) ("id" ("wa" "ph" ("wn" "ps")))) ("olc" ("wa" "ph" ("wn" "ps")) ("wa" "ph" ("wn" "ph")))))) ;; Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.) (theorem "pm5_21nd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("pm5_21nd_1" ("wi" ("wa" "ph" "ps") "th")) ("pm5_21nd_2" ("wi" ("wa" "ph" "ch") "th")) ("pm5_21nd_3" ("wi" "th" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" "ps" "ch")) ("pm5_21ndd" "ph" "th" "ps" "ch" ("ex" "ph" "ps" "th" "pm5_21nd_1") ("ex" "ph" "ch" "th" "pm5_21nd_2") ("a1i" ("wi" "th" ("wb" "ps" "ch")) "ph" "pm5_21nd_3"))) ;; Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_35" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" ("wi" "ph" "ps") ("wi" "ph" "ch")) ("wi" "ph" ("wb" "ps" "ch"))) ("pm5_74rd" ("wa" ("wi" "ph" "ps") ("wi" "ph" "ch")) "ph" "ps" "ch" ("pm5_1" ("wi" "ph" "ps") ("wi" "ph" "ch")))) ;; Theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 7-Nov-2013.) (theorem "pm5_54" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wo" ("wb" ("wa" "ph" "ps") "ph") ("wb" ("wa" "ph" "ps") "ps")) ("orri" ("wb" ("wa" "ph" "ps") "ph") ("wb" ("wa" "ph" "ps") "ps") ("pm5_21ni" ("wa" "ph" "ps") ("wb" ("wa" "ph" "ps") "ph") "ps" ("adantl" "ps" ("wb" ("wa" "ph" "ps") "ph") "ph" ("bicomd" "ps" "ph" ("wa" "ph" "ps") ("iba" "ps" "ph"))) ("bicomd" "ps" "ph" ("wa" "ph" "ps") ("iba" "ps" "ph"))))) ;; Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.) (theorem "baib" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("baib_1" ("wb" "ph" ("wa" "ps" "ch")))) (for) ("wi" "ps" ("wb" "ph" "ch")) ("syl6rbbr" "ps" "ch" ("wa" "ps" "ch") "ph" ("ibar" "ps" "ch") "baib_1")) ;; Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.) (theorem "baibr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("baib_1" ("wb" "ph" ("wa" "ps" "ch")))) (for) ("wi" "ps" ("wb" "ch" "ph")) ("bicomd" "ps" "ph" "ch" ("baib" "ph" "ps" "ch" "baib_1"))) ;; Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) (Proof shortened by Wolf Lammen, 19-Jan-2020.) (theorem "rbaibr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("baib_1" ("wb" "ph" ("wa" "ps" "ch")))) (for) ("wi" "ch" ("wb" "ps" "ph")) ("syl6bbr" "ch" "ps" ("wa" "ps" "ch") "ph" ("iba" "ch" "ps") "baib_1")) ;; Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) (Proof shortened by Wolf Lammen, 19-Jan-2020.) (theorem "rbaib" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("baib_1" ("wb" "ph" ("wa" "ps" "ch")))) (for) ("wi" "ch" ("wb" "ph" "ps")) ("bicomd" "ch" "ps" "ph" ("rbaibr" "ph" "ps" "ch" "baib_1"))) ;; Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) (theorem "baibd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("baibd_1" ("wi" "ph" ("wb" "ps" ("wa" "ch" "th"))))) (for) ("wi" ("wa" "ph" "ch") ("wb" "ps" "th")) ("sylan9bb" "ph" "ps" ("wa" "ch" "th") "ch" "th" "baibd_1" ("bicomd" "ch" "th" ("wa" "ch" "th") ("ibar" "ch" "th")))) ;; Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) (theorem "rbaibd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("baibd_1" ("wi" "ph" ("wb" "ps" ("wa" "ch" "th"))))) (for) ("wi" ("wa" "ph" "th") ("wb" "ps" "ch")) ("sylan9bb" "ph" "ps" ("wa" "ch" "th") "th" "ch" "baibd_1" ("bicomd" "th" "ch" ("wa" "ch" "th") ("iba" "th" "ch")))) ;; Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_44" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wb" ("wi" "ph" "ch") ("wi" "ph" ("wa" "ps" "ch")))) ("baibr" ("wi" "ph" ("wa" "ps" "ch")) ("wi" "ph" "ps") ("wi" "ph" "ch") ("jcab" "ph" "ps" "ch"))) ;; Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of [WhiteheadRussell] p. 125. (Contributed by NM, 8-Jun-1994.) (theorem "pm5_6" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wi" ("wa" "ph" ("wn" "ps")) "ch") ("wi" "ph" ("wo" "ps" "ch"))) ("bitr4i" ("wi" ("wa" "ph" ("wn" "ps")) "ch") ("wi" "ph" ("wi" ("wn" "ps") "ch")) ("wi" "ph" ("wo" "ps" "ch")) ("impexp" "ph" ("wn" "ps") "ch") ("imbi2i" ("wo" "ps" "ch") ("wi" ("wn" "ps") "ch") "ph" ("df_or" "ps" "ch")))) ;; Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.) (theorem "orcanai" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("orcanai_1" ("wi" "ph" ("wo" "ps" "ch")))) (for) ("wi" ("wa" "ph" ("wn" "ps")) "ch") ("imp" "ph" ("wn" "ps") "ch" ("ord" "ph" "ps" "ch" "orcanai_1"))) ;; Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (theorem "mpbiran" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpbiran_1" "ps") ("mpbiran_2" ("wb" "ph" ("wa" "ps" "ch")))) (for) ("wb" "ph" "ch") ("bitr4i" "ph" ("wa" "ps" "ch") "ch" "mpbiran_2" ("biantrur" "ps" "ch" "mpbiran_1"))) ;; Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (theorem "mpbiran2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpbiran2_1" "ch") ("mpbiran2_2" ("wb" "ph" ("wa" "ps" "ch")))) (for) ("wb" "ph" "ps") ("bitr4i" "ph" ("wa" "ps" "ch") "ps" "mpbiran2_2" ("biantru" "ch" "ps" "mpbiran2_1"))) ;; Detach a conjunction of truths in a biconditional. (Contributed by NM, 10-May-2005.) (theorem "mpbir2an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("mpbir2an_1" "ps") ("mpbir2an_2" "ch") ("mpbiran2an_1" ("wb" "ph" ("wa" "ps" "ch")))) (for) "ph" ("mpbir" "ph" "ch" "mpbir2an_2" ("mpbiran" "ph" "ps" "ch" "mpbir2an_1" "mpbiran2an_1"))) ;; Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) (theorem "mpbi2and" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpbi2and_1" ("wi" "ph" "ps")) ("mpbi2and_2" ("wi" "ph" "ch")) ("mpbi2and_3" ("wi" "ph" ("wb" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" "th") ("mpbid" "ph" ("wa" "ps" "ch") "th" ("jca" "ph" "ps" "ch" "mpbi2and_1" "mpbi2and_2") "mpbi2and_3")) ;; Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) (theorem "mpbir2and" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpbir2and_1" ("wi" "ph" "ch")) ("mpbir2and_2" ("wi" "ph" "th")) ("mpbir2and_3" ("wi" "ph" ("wb" "ps" ("wa" "ch" "th"))))) (for) ("wi" "ph" "ps") ("mpbird" "ph" "ps" ("wa" "ch" "th") ("jca" "ph" "ch" "th" "mpbir2and_1" "mpbir2and_2") "mpbir2and_3")) ;; Theorem *5.62 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 21-Jun-2005.) (theorem "pm5_62" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wo" ("wa" "ph" "ps") ("wn" "ps")) ("wo" "ph" ("wn" "ps"))) ("mpbiran2" ("wo" ("wa" "ph" "ps") ("wn" "ps")) ("wo" "ph" ("wn" "ps")) ("wo" "ps" ("wn" "ps")) ("exmid" "ps") ("ordir" "ph" "ps" ("wn" "ps")))) ;; Theorem *5.63 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 25-Dec-2012.) (theorem "pm5_63" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wo" "ph" "ps") ("wo" "ph" ("wa" ("wn" "ph") "ps"))) ("bicomi" ("wo" "ph" ("wa" ("wn" "ph") "ps")) ("wo" "ph" "ps") ("mpbiran" ("wo" "ph" ("wa" ("wn" "ph") "ps")) ("wo" "ph" ("wn" "ph")) ("wo" "ph" "ps") ("exmid" "ph") ("ordi" "ph" ("wn" "ph") "ps")))) ;; Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.) (theorem "intnan" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("intnan_1" ("wn" "ph"))) (for) ("wn" ("wa" "ps" "ph")) ("mto" ("wa" "ps" "ph") "ph" "intnan_1" ("simpr" "ps" "ph"))) ;; Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.) (theorem "intnanr" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("intnan_1" ("wn" "ph"))) (for) ("wn" ("wa" "ph" "ps")) ("mto" ("wa" "ph" "ps") "ph" "intnan_1" ("simpl" "ph" "ps"))) ;; Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.) (theorem "intnand" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("intnand_1" ("wi" "ph" ("wn" "ps")))) (for) ("wi" "ph" ("wn" ("wa" "ch" "ps"))) ("nsyl" "ph" "ps" ("wa" "ch" "ps") "intnand_1" ("simpr" "ch" "ps"))) ;; Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.) (theorem "intnanrd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("intnand_1" ("wi" "ph" ("wn" "ps")))) (for) ("wi" "ph" ("wn" ("wa" "ps" "ch"))) ("nsyl" "ph" "ps" ("wa" "ps" "ch") "intnand_1" ("simpl" "ps" "ch"))) ;; Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.) (theorem "niabn" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("niabn_1" "ph")) (for) ("wi" ("wn" "ps") ("wb" ("wa" "ch" "ps") ("wn" "ph"))) ("pm5_21ni" ("wa" "ch" "ps") "ps" ("wn" "ph") ("simpr" "ch" "ps") ("pm2_24i" "ph" "ps" "niabn_1"))) ;; Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.) (theorem "ninba" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ninba_1" "ph")) (for) ("wi" ("wn" "ps") ("wb" ("wn" "ph") ("wa" "ch" "ps"))) ("bicomd" ("wn" "ps") ("wa" "ch" "ps") ("wn" "ph") ("niabn" "ph" "ps" "ch" "ninba_1"))) ;; A wff conjoined with falsehood is false. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) (theorem "bianfi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("bianfi_1" ("wn" "ph"))) (for) ("wb" "ph" ("wa" "ps" "ph")) ("_2false" "ph" ("wa" "ps" "ph") "bianfi_1" ("intnan" "ph" "ps" "bianfi_1"))) ;; A wff conjoined with falsehood is false. (Contributed by NM, 27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.) (theorem "bianfd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("bianfd_1" ("wi" "ph" ("wn" "ps")))) (for) ("wi" "ph" ("wb" "ps" ("wa" "ps" "ch"))) ("_2falsed" "ph" "ps" ("wa" "ps" "ch") "bianfd_1" ("intnanrd" "ph" "ps" "ch" "bianfd_1"))) ;; Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) (theorem "pm4_43" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" "ph" ("wa" ("wo" "ph" "ps") ("wo" "ph" ("wn" "ps")))) ("bitri" "ph" ("wo" "ph" ("wa" "ps" ("wn" "ps"))) ("wa" ("wo" "ph" "ps") ("wo" "ph" ("wn" "ps"))) ("biorfi" ("wa" "ps" ("wn" "ps")) "ph" ("pm3_24" "ps")) ("ordi" "ph" "ps" ("wn" "ps")))) ;; Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_82" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" ("wi" "ph" "ps") ("wi" "ph" ("wn" "ps"))) ("wn" "ph")) ("impbii" ("wa" ("wi" "ph" "ps") ("wi" "ph" ("wn" "ps"))) ("wn" "ph") ("imp" ("wi" "ph" "ps") ("wi" "ph" ("wn" "ps")) ("wn" "ph") ("pm2_65" "ph" "ps")) ("jca" ("wn" "ph") ("wi" "ph" "ps") ("wi" "ph" ("wn" "ps")) ("pm2_21" "ph" "ps") ("pm2_21" "ph" ("wn" "ps"))))) ;; Theorem *4.83 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (theorem "pm4_83" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ps")) "ps") ("bitr2i" "ps" ("wi" ("wo" "ph" ("wn" "ph")) "ps") ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ps")) ("a1bi" ("wo" "ph" ("wn" "ph")) "ps" ("exmid" "ph")) ("jaob" "ph" "ps" ("wn" "ph")))) ;; Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Nov-2012.) (theorem "pclem6" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wb" "ph" ("wa" "ps" ("wn" "ph"))) ("wn" "ps")) ("con2i" "ps" ("wb" "ph" ("wa" "ps" ("wn" "ph"))) ("sylib" "ps" ("wb" ("wn" "ph") ("wa" "ps" ("wn" "ph"))) ("wn" ("wb" "ph" ("wa" "ps" ("wn" "ph")))) ("ibar" "ps" ("wn" "ph")) ("nbbn" "ph" ("wa" "ps" ("wn" "ph")))))) ;; A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.) (theorem "biantr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" ("wb" "ph" "ps") ("wb" "ch" "ps")) ("wb" "ph" "ch")) ("biimparc" ("wb" "ch" "ps") ("wb" "ph" "ch") ("wb" "ph" "ps") ("bibi2d" ("wb" "ch" "ps") "ch" "ps" "ph" ("id" ("wb" "ch" "ps"))))) ;; Disjunction distributes over the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), ~ http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384 . (Contributed by NM, 8-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Feb-2013.) (theorem "orbidi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" "ph" ("wb" "ps" "ch")) ("wb" ("wo" "ph" "ps") ("wo" "ph" "ch"))) ("_3bitr4i" ("wi" ("wn" "ph") ("wb" "ps" "ch")) ("wb" ("wi" ("wn" "ph") "ps") ("wi" ("wn" "ph") "ch")) ("wo" "ph" ("wb" "ps" "ch")) ("wb" ("wo" "ph" "ps") ("wo" "ph" "ch")) ("pm5_74" ("wn" "ph") "ps" "ch") ("df_or" "ph" ("wb" "ps" "ch")) ("bibi12i" ("wo" "ph" "ps") ("wi" ("wn" "ph") "ps") ("wo" "ph" "ch") ("wi" ("wn" "ph") "ch") ("df_or" "ph" "ps") ("df_or" "ph" "ch")))) ;; Lukasiewicz's shortest axiom for equivalential calculus. Storrs McCall, ed., _Polish Logic 1920-1939_ (Oxford, 1967), p. 96. (Contributed by NM, 10-Jan-2005.) (theorem "biluk" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wb" "ph" "ps") ("wb" ("wb" "ch" "ps") ("wb" "ph" "ch"))) ("bitr4i" ("wb" "ph" "ps") ("wb" "ch" ("wb" "ps" ("wb" "ph" "ch"))) ("wb" ("wb" "ch" "ps") ("wb" "ph" "ch")) ("mpbi" ("wb" ("wb" ("wb" "ph" "ps") "ch") ("wb" "ps" ("wb" "ph" "ch"))) ("wb" ("wb" "ph" "ps") ("wb" "ch" ("wb" "ps" ("wb" "ph" "ch")))) ("bitri" ("wb" ("wb" "ph" "ps") "ch") ("wb" ("wb" "ps" "ph") "ch") ("wb" "ps" ("wb" "ph" "ch")) ("bibi1i" ("wb" "ph" "ps") ("wb" "ps" "ph") "ch" ("bicom" "ph" "ps")) ("biass" "ps" "ph" "ch")) ("biass" ("wb" "ph" "ps") "ch" ("wb" "ps" ("wb" "ph" "ch")))) ("biass" "ch" "ps" ("wb" "ph" "ch")))) ;; Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to ~ orbidi . (Contributed by Roy F. Longton, 21-Jun-2005.) (theorem "pm5_7" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wb" ("wo" "ph" "ch") ("wo" "ps" "ch")) ("wo" "ch" ("wb" "ph" "ps"))) ("bitr2i" ("wo" "ch" ("wb" "ph" "ps")) ("wb" ("wo" "ch" "ph") ("wo" "ch" "ps")) ("wb" ("wo" "ph" "ch") ("wo" "ps" "ch")) ("orbidi" "ch" "ph" "ps") ("bibi12i" ("wo" "ch" "ph") ("wo" "ph" "ch") ("wo" "ch" "ps") ("wo" "ps" "ch") ("orcom" "ch" "ph") ("orcom" "ch" "ps")))) ;; Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.) (theorem "bigolden" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" ("wa" "ph" "ps") "ph") ("wb" "ps" ("wo" "ph" "ps"))) ("_3bitr3ri" ("wi" "ph" "ps") ("wb" "ph" ("wa" "ph" "ps")) ("wb" "ps" ("wo" "ph" "ps")) ("wb" ("wa" "ph" "ps") "ph") ("pm4_71" "ph" "ps") ("pm4_72" "ph" "ps") ("bicom" "ph" ("wa" "ph" "ps")))) ;; Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.) (theorem "pm5_71" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ps" ("wn" "ch")) ("wb" ("wa" ("wo" "ph" "ps") "ch") ("wa" "ph" "ch"))) ("ja" "ps" ("wn" "ch") ("wb" ("wa" ("wo" "ph" "ps") "ch") ("wa" "ph" "ch")) ("anbi1d" ("wn" "ps") ("wo" "ph" "ps") "ph" "ch" ("impbid1" ("wn" "ps") ("wo" "ph" "ps") "ph" ("orel2" "ps" "ph") ("orc" "ph" "ps"))) ("pm5_32rd" ("wn" "ch") "ch" ("wo" "ph" "ps") "ph" ("pm2_21" "ch" ("wb" ("wo" "ph" "ps") "ph"))))) ;; Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.) (Proof shortened by Kyle Wyonch, 12-Feb-2021.) (theorem "pm5_75" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" ("wi" "ch" ("wn" "ps")) ("wb" "ph" ("wo" "ps" "ch"))) ("wb" ("wa" "ph" ("wn" "ps")) "ch")) ("sylan9bbr" ("wb" "ph" ("wo" "ps" "ch")) ("wa" "ph" ("wn" "ps")) ("wa" "ch" ("wn" "ps")) ("wi" "ch" ("wn" "ps")) "ch" ("syl6bb" ("wb" "ph" ("wo" "ps" "ch")) ("wa" "ph" ("wn" "ps")) ("wa" ("wo" "ps" "ch") ("wn" "ps")) ("wa" "ch" ("wn" "ps")) ("anbi1" "ph" ("wo" "ps" "ch") ("wn" "ps")) ("pm5_32ri" ("wn" "ps") ("wo" "ps" "ch") "ch" ("bicomd" ("wn" "ps") "ch" ("wo" "ps" "ch") ("biorf" "ps" "ch")))) ("bicomd" ("wi" "ch" ("wn" "ps")) "ch" ("wa" "ch" ("wn" "ps")) ("biimpi" ("wi" "ch" ("wn" "ps")) ("wb" "ch" ("wa" "ch" ("wn" "ps"))) ("pm4_71" "ch" ("wn" "ps")))))) ;; Removal of conjunct from one side of an equivalence. (Contributed by NM, 21-Jun-1993.) (theorem "bimsc1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" ("wi" "ph" "ps") ("wb" "ch" ("wa" "ps" "ph"))) ("wb" "ch" "ph")) ("biimpa" ("wi" "ph" "ps") ("wb" "ch" ("wa" "ps" "ph")) ("wb" "ch" "ph") ("bibi2d" ("wi" "ph" "ps") ("wa" "ps" "ph") "ph" "ch" ("impbid2" ("wi" "ph" "ps") ("wa" "ps" "ph") "ph" ("simpr" "ps" "ph") ("ancr" "ph" "ps"))))) ;; Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Dec-2012.) (theorem "ecase2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ecase2d_1" ("wi" "ph" "ps")) ("ecase2d_2" ("wi" "ph" ("wn" ("wa" "ps" "ch")))) ("ecase2d_3" ("wi" "ph" ("wn" ("wa" "ps" "th")))) ("ecase2d_4" ("wi" "ph" ("wo" "ta" ("wo" "ch" "th"))))) (for) ("wi" "ph" "ta") ("mpjaod" "ph" "ta" "ta" ("wo" "ch" "th") ("idd" "ph" "ta") ("jaod" "ph" "ch" "ta" "th" ("mpand" "ph" "ps" "ch" "ta" "ecase2d_1" ("pm2_21d" "ph" ("wa" "ps" "ch") "ta" "ecase2d_2")) ("mpand" "ph" "ps" "th" "ta" "ecase2d_1" ("pm2_21d" "ph" ("wa" "ps" "th") "ta" "ecase2d_3"))) "ecase2d_4")) ;; Inference for elimination by cases. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) (theorem "ecase3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ecase3_1" ("wi" "ph" "ch")) ("ecase3_2" ("wi" "ps" "ch")) ("ecase3_3" ("wi" ("wn" ("wo" "ph" "ps")) "ch"))) (for) "ch" ("pm2_61i" ("wo" "ph" "ps") "ch" ("jaoi" "ph" "ch" "ps" "ecase3_1" "ecase3_2") "ecase3_3")) ;; Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.) (theorem "ecase" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("ecase_1" ("wi" ("wn" "ph") "ch")) ("ecase_2" ("wi" ("wn" "ps") "ch")) ("ecase_3" ("wi" ("wa" "ph" "ps") "ch"))) (for) "ch" ("pm2_61nii" "ph" "ps" "ch" ("ex" "ph" "ps" "ch" "ecase_3") "ecase_1" "ecase_2")) ;; Deduction for elimination by cases. (Contributed by NM, 2-May-1996.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "ecase3d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("ecase3d_1" ("wi" "ph" ("wi" "ps" "th"))) ("ecase3d_2" ("wi" "ph" ("wi" "ch" "th"))) ("ecase3d_3" ("wi" "ph" ("wi" ("wn" ("wo" "ps" "ch")) "th")))) (for) ("wi" "ph" "th") ("pm2_61d" "ph" ("wo" "ps" "ch") "th" ("jaod" "ph" "ps" "th" "ch" "ecase3d_1" "ecase3d_2") "ecase3d_3")) ;; Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012.) (theorem "ecased" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("ecased_1" ("wi" "ph" ("wi" ("wn" "ps") "th"))) ("ecased_2" ("wi" "ph" ("wi" ("wn" "ch") "th"))) ("ecased_3" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" "ph" "th") ("ecase3d" "ph" ("wn" "ps") ("wn" "ch") "th" "ecased_1" "ecased_2" ("syl5" ("wn" ("wo" ("wn" "ps") ("wn" "ch"))) ("wa" "ps" "ch") "ph" "th" ("pm3_11" "ps" "ch") "ecased_3"))) ;; Deduction for elimination by cases. (Contributed by NM, 24-May-2013.) (theorem "ecase3ad" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("ecase3ad_1" ("wi" "ph" ("wi" "ps" "th"))) ("ecase3ad_2" ("wi" "ph" ("wi" "ch" "th"))) ("ecase3ad_3" ("wi" "ph" ("wi" ("wa" ("wn" "ps") ("wn" "ch")) "th")))) (for) ("wi" "ph" "th") ("ecased" "ph" ("wn" "ps") ("wn" "ch") "th" ("syl5" ("wn" ("wn" "ps")) "ps" "ph" "th" ("notnotr" "ps") "ecase3ad_1") ("syl5" ("wn" ("wn" "ch")) "ch" "ph" "th" ("notnotr" "ch") "ecase3ad_2") "ecase3ad_3")) ;; Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.) (theorem "ccase" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ccase_1" ("wi" ("wa" "ph" "ps") "ta")) ("ccase_2" ("wi" ("wa" "ch" "ps") "ta")) ("ccase_3" ("wi" ("wa" "ph" "th") "ta")) ("ccase_4" ("wi" ("wa" "ch" "th") "ta"))) (for) ("wi" ("wa" ("wo" "ph" "ch") ("wo" "ps" "th")) "ta") ("jaodan" ("wo" "ph" "ch") "ps" "ta" "th" ("jaoian" "ph" "ps" "ta" "ch" "ccase_1" "ccase_2") ("jaoian" "ph" "th" "ta" "ch" "ccase_3" "ccase_4"))) ;; Deduction for combining cases. (Contributed by NM, 9-May-2004.) (theorem "ccased" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ccased_1" ("wi" "ph" ("wi" ("wa" "ps" "ch") "et"))) ("ccased_2" ("wi" "ph" ("wi" ("wa" "th" "ch") "et"))) ("ccased_3" ("wi" "ph" ("wi" ("wa" "ps" "ta") "et"))) ("ccased_4" ("wi" "ph" ("wi" ("wa" "th" "ta") "et")))) (for) ("wi" "ph" ("wi" ("wa" ("wo" "ps" "th") ("wo" "ch" "ta")) "et")) ("com12" ("wa" ("wo" "ps" "th") ("wo" "ch" "ta")) "ph" "et" ("ccase" "ps" "ch" "th" "ta" ("wi" "ph" "et") ("com12" "ph" ("wa" "ps" "ch") "et" "ccased_1") ("com12" "ph" ("wa" "th" "ch") "et" "ccased_2") ("com12" "ph" ("wa" "ps" "ta") "et" "ccased_3") ("com12" "ph" ("wa" "th" "ta") "et" "ccased_4")))) ;; Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (theorem "ccase2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ccase2_1" ("wi" ("wa" "ph" "ps") "ta")) ("ccase2_2" ("wi" "ch" "ta")) ("ccase2_3" ("wi" "th" "ta"))) (for) ("wi" ("wa" ("wo" "ph" "ch") ("wo" "ps" "th")) "ta") ("ccase" "ph" "ps" "ch" "th" "ta" "ccase2_1" ("adantr" "ch" "ta" "ps" "ccase2_2") ("adantl" "th" "ta" "ph" "ccase2_3") ("adantl" "th" "ta" "ch" "ccase2_3"))) ;; Inference eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 25-Oct-2003.) (theorem "_4cases" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_4cases_1" ("wi" ("wa" "ph" "ps") "ch")) ("_4cases_2" ("wi" ("wa" "ph" ("wn" "ps")) "ch")) ("_4cases_3" ("wi" ("wa" ("wn" "ph") "ps") "ch")) ("_4cases_4" ("wi" ("wa" ("wn" "ph") ("wn" "ps")) "ch"))) (for) "ch" ("pm2_61i" "ps" "ch" ("pm2_61ian" "ph" "ps" "ch" "_4cases_1" "_4cases_3") ("pm2_61ian" "ph" ("wn" "ps") "ch" "_4cases_2" "_4cases_4"))) ;; Deduction eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 19-Mar-2013.) (theorem "_4casesdan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_4casesdan_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th")) ("_4casesdan_2" ("wi" ("wa" "ph" ("wa" "ps" ("wn" "ch"))) "th")) ("_4casesdan_3" ("wi" ("wa" "ph" ("wa" ("wn" "ps") "ch")) "th")) ("_4casesdan_4" ("wi" ("wa" "ph" ("wa" ("wn" "ps") ("wn" "ch"))) "th"))) (for) ("wi" "ph" "th") ("_4cases" "ps" "ch" ("wi" "ph" "th") ("expcom" "ph" ("wa" "ps" "ch") "th" "_4casesdan_1") ("expcom" "ph" ("wa" "ps" ("wn" "ch")) "th" "_4casesdan_2") ("expcom" "ph" ("wa" ("wn" "ps") "ch") "th" "_4casesdan_3") ("expcom" "ph" ("wa" ("wn" "ps") ("wn" "ch")) "th" "_4casesdan_4"))) ;; Case disjunction according to the value of ` ph ` . (Contributed by NM, 25-Apr-2019.) (theorem "cases" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("cases_1" ("wi" "ph" ("wb" "ps" "ch"))) ("cases_2" ("wi" ("wn" "ph") ("wb" "ps" "th")))) (for) ("wb" "ps" ("wo" ("wa" "ph" "ch") ("wa" ("wn" "ph") "th"))) ("_3bitri" "ps" ("wa" ("wo" "ph" ("wn" "ph")) "ps") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ps")) ("wo" ("wa" "ph" "ch") ("wa" ("wn" "ph") "th")) ("biantrur" ("wo" "ph" ("wn" "ph")) "ps" ("exmid" "ph")) ("andir" "ph" ("wn" "ph") "ps") ("orbi12i" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" ("wn" "ph") "ps") ("wa" ("wn" "ph") "th") ("pm5_32i" "ph" "ps" "ch" "cases_1") ("pm5_32i" ("wn" "ph") "ps" "th" "cases_2")))) ;; Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (theorem "dedlem0a" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" "ph" ("wb" "ps" ("wi" ("wi" "ch" "ph") ("wa" "ps" "ph")))) ("bitrd" "ph" "ps" ("wa" "ps" "ph") ("wi" ("wi" "ch" "ph") ("wa" "ps" "ph")) ("iba" "ph" "ps") ("syl" "ph" ("wi" "ch" "ph") ("wb" ("wa" "ps" "ph") ("wi" ("wi" "ch" "ph") ("wa" "ps" "ph"))) ("ax_1" "ph" "ch") ("biimt" ("wi" "ch" "ph") ("wa" "ps" "ph"))))) ;; Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) (theorem "dedlem0b" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wb" "ps" ("wi" ("wi" "ps" "ph") ("wa" "ch" "ph")))) ("impbid" ("wn" "ph") "ps" ("wi" ("wi" "ps" "ph") ("wa" "ch" "ph")) ("com23" ("wn" "ph") ("wi" "ps" "ph") "ps" ("wa" "ch" "ph") ("imim2d" ("wn" "ph") "ph" ("wa" "ch" "ph") "ps" ("pm2_21" "ph" ("wa" "ch" "ph")))) ("com12" ("wi" ("wi" "ps" "ph") ("wa" "ch" "ph")) ("wn" "ph") "ps" ("con1d" ("wi" ("wi" "ps" "ph") ("wa" "ch" "ph")) "ps" "ph" ("imim12i" ("wn" "ps") ("wi" "ps" "ph") ("wa" "ch" "ph") "ph" ("pm2_21" "ps" "ph") ("simpr" "ch" "ph")))))) ;; Lemma for weak deduction theorem. See also ~ ifptru . (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "dedlema" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" "ph" ("wb" "ps" ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph"))))) ("impbid" "ph" "ps" ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph"))) ("expcom" "ps" "ph" ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph"))) ("orc" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph")))) ("jaod" "ph" ("wa" "ps" "ph") "ps" ("wa" "ch" ("wn" "ph")) ("a1i" ("wi" ("wa" "ps" "ph") "ps") "ph" ("simpl" "ps" "ph")) ("adantld" "ph" ("wn" "ph") "ps" "ch" ("pm2_24" "ph" "ps"))))) ;; Lemma for weak deduction theorem. See also ~ ifpfal . (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.) (theorem "dedlemb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wb" "ch" ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph"))))) ("impbid" ("wn" "ph") "ch" ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph"))) ("expcom" "ch" ("wn" "ph") ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph"))) ("olc" ("wa" "ch" ("wn" "ph")) ("wa" "ps" "ph"))) ("jaod" ("wn" "ph") ("wa" "ps" "ph") "ch" ("wa" "ch" ("wn" "ph")) ("adantld" ("wn" "ph") "ph" "ch" "ps" ("pm2_21" "ph" "ch")) ("a1i" ("wi" ("wa" "ch" ("wn" "ph")) "ch") ("wn" "ph") ("simpl" "ch" ("wn" "ph")))))) ;; Case disjunction according to the value of ` ph ` . (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Wolf Lammen, 28-Feb-2022.) (theorem "cases2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch"))) ("_3bitr4ri" ("wa" ("wi" "ph" ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph")))) ("wi" ("wn" "ph") ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph"))))) ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph"))) ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch")) ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("pm4_83" "ph" ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph")))) ("anbi12i" ("wi" "ph" "ps") ("wi" "ph" ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph")))) ("wi" ("wn" "ph") "ch") ("wi" ("wn" "ph") ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph")))) ("pm5_74i" "ph" "ps" ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph"))) ("dedlema" "ph" "ps" "ch")) ("pm5_74i" ("wn" "ph") "ch" ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wn" "ph"))) ("dedlemb" "ph" "ps" "ch"))) ("orbi12i" ("wa" "ph" "ps") ("wa" "ps" "ph") ("wa" ("wn" "ph") "ch") ("wa" "ch" ("wn" "ph")) ("ancom" "ph" "ps") ("ancom" ("wn" "ph") "ch")))) ;; Alternate proof of ~ cases2 , not using ~ dedlema or ~ dedlemb . (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Wolf Lammen, 2-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "cases2ALT" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch"))) ("impbii" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch")) ("jaoi" ("wa" "ph" "ps") ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch")) ("wa" ("wn" "ph") "ch") ("jca" ("wa" "ph" "ps") ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch") ("pm3_4" "ph" "ps") ("adantr" "ph" ("wi" ("wn" "ph") "ch") "ps" ("pm2_24" "ph" "ch"))) ("jca" ("wa" ("wn" "ph") "ch") ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch") ("adantr" ("wn" "ph") ("wi" "ph" "ps") "ch" ("pm2_21" "ph" "ps")) ("pm3_4" ("wn" "ph") "ch"))) ("pm2_61ian" "ph" ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch")) ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("adantrr" "ph" ("wi" "ph" "ps") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wi" ("wn" "ph") "ch") ("orcd" ("wa" "ph" ("wi" "ph" "ps")) ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch") ("imdistani" "ph" ("wi" "ph" "ps") "ps" ("pm2_27" "ph" "ps")))) ("adantrl" ("wn" "ph") ("wi" ("wn" "ph") "ch") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wi" "ph" "ps") ("olcd" ("wa" ("wn" "ph") ("wi" ("wn" "ph") "ch")) ("wa" ("wn" "ph") "ch") ("wa" "ph" "ps") ("imdistani" ("wn" "ph") ("wi" ("wn" "ph") "ch") "ch" ("pm2_27" ("wn" "ph") "ch"))))))) ;; An alternate definition of the biconditional. Theorem *5.23 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) (Proof shortened by NM, 29-Oct-2021.) (theorem "dfbi3" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" "ph" "ps") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps")))) ("_3bitr4i" ("wa" ("wi" "ph" "ps") ("wi" "ps" "ph")) ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") ("wn" "ps"))) ("wb" "ph" "ps") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps"))) ("anbi2i" ("wi" "ps" "ph") ("wi" ("wn" "ph") ("wn" "ps")) ("wi" "ph" "ps") ("con34b" "ps" "ph")) ("dfbi2" "ph" "ps") ("cases2" "ph" "ps" ("wn" "ps")))) ;; Obsolete proof of ~ dfbi3 as of 29-Oct-2021. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) (New usage is discouraged.) (Proof modification is discouraged.) (theorem "dfbi3OLD" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" "ph" "ps") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps")))) ("_3bitr4i" ("wn" ("wb" "ph" ("wn" "ps"))) ("wo" ("wa" "ph" ("wn" ("wn" "ps"))) ("wa" ("wn" "ps") ("wn" "ph"))) ("wb" "ph" "ps") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps"))) ("xor" "ph" ("wn" "ps")) ("pm5_18" "ph" "ps") ("orbi12i" ("wa" "ph" "ps") ("wa" "ph" ("wn" ("wn" "ps"))) ("wa" ("wn" "ph") ("wn" "ps")) ("wa" ("wn" "ps") ("wn" "ph")) ("anbi2i" "ps" ("wn" ("wn" "ps")) "ph" ("notnotb" "ps")) ("ancom" ("wn" "ph") ("wn" "ps"))))) ;; Theorem *5.24 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (theorem "pm5_24" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wn" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps")))) ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph")))) ("xchnxbi" ("wb" "ph" "ps") ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph"))) ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps"))) ("xor" "ph" "ps") ("dfbi3" "ph" "ps"))) ;; The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see ~ exmid ). (Contributed by David Abernethy, 28-Jan-2014.) (Proof shortened by NM, 29-Oct-2021.) (theorem "_4exmid" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wo" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps"))) ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph")))) ("orri" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps"))) ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph"))) ("biimpi" ("wn" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps")))) ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph"))) ("pm5_24" "ph" "ps")))) ;; Obsolete proof of ~ 4exmid as of 29-Oct-2021. (Contributed by David Abernethy, 28-Jan-2014.) (New usage is discouraged.) (Proof modification is discouraged.) (theorem "_4exmidOLD" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wo" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps"))) ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph")))) ("mpbi" ("wo" ("wb" "ph" "ps") ("wn" ("wb" "ph" "ps"))) ("wo" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps"))) ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph")))) ("exmid" ("wb" "ph" "ps")) ("orbi12i" ("wb" "ph" "ps") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps"))) ("wn" ("wb" "ph" "ps")) ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph"))) ("dfbi3" "ph" "ps") ("xor" "ph" "ps")))) ;; The consensus theorem. This theorem and its dual (with ` \/ ` and ` /\ ` interchanged) are commonly used in computer logic design to eliminate redundant terms from Boolean expressions. Specifically, we prove that the term ` ( ps /\ ch ) ` on the left-hand side is redundant. (Contributed by NM, 16-May-2003.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 20-Jan-2013.) (theorem "consensus" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wo" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wa" "ps" "ch")) ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch"))) ("impbii" ("wo" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wa" "ps" "ch")) ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("jaoi" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wa" "ps" "ch") ("id" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch"))) ("pm2_61ian" "ph" ("wa" "ps" "ch") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("adantrr" "ph" "ps" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) "ch" ("orc" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch"))) ("adantrl" ("wn" "ph") "ch" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) "ps" ("olc" ("wa" ("wn" "ph") "ch") ("wa" "ph" "ps"))))) ("orc" ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wa" "ps" "ch")))) ;; Theorem *4.42 of [WhiteheadRussell] p. 119. See also ~ ifpid . (Contributed by Roy F. Longton, 21-Jun-2005.) (theorem "pm4_42" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" "ph" ("wo" ("wa" "ph" "ps") ("wa" "ph" ("wn" "ps")))) ("pm2_61i" "ps" ("wb" "ph" ("wo" ("wa" "ph" "ps") ("wa" "ph" ("wn" "ps")))) ("dedlema" "ps" "ph" "ph") ("dedlemb" "ps" "ph" "ph"))) ;; A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) (theorem "prlem1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("prlem1_1" ("wi" "ph" ("wb" "et" "ch"))) ("prlem1_2" ("wi" "ps" ("wn" "th")))) (for) ("wi" "ph" ("wi" "ps" ("wi" ("wo" ("wa" "ps" "ch") ("wa" "th" "ta")) "et"))) ("ex" "ph" "ps" ("wi" ("wo" ("wa" "ps" "ch") ("wa" "th" "ta")) "et") ("jaao" "ph" ("wa" "ps" "ch") "et" "ps" ("wa" "th" "ta") ("adantld" "ph" "ch" "et" "ps" ("biimprd" "ph" "et" "ch" "prlem1_1")) ("adantrd" "ps" "th" "et" "ta" ("pm2_21d" "ps" "th" "et" "prlem1_2"))))) ;; A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 9-Dec-2012.) (theorem "prlem2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wo" ("wa" "ph" "ps") ("wa" "ch" "th")) ("wa" ("wo" "ph" "ch") ("wo" ("wa" "ph" "ps") ("wa" "ch" "th")))) ("pm4_71ri" ("wo" ("wa" "ph" "ps") ("wa" "ch" "th")) ("wo" "ph" "ch") ("orim12i" ("wa" "ph" "ps") "ph" ("wa" "ch" "th") "ch" ("simpl" "ph" "ps") ("simpl" "ch" "th")))) ;; A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) (theorem "oplem1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("oplem1_1" ("wi" "ph" ("wo" "ps" "ch"))) ("oplem1_2" ("wi" "ph" ("wo" "th" "ta"))) ("oplem1_3" ("wb" "ps" "th")) ("oplem1_4" ("wi" "ch" ("wb" "th" "ta")))) (for) ("wi" "ph" "ps") ("sylibr" "ph" "th" "ps" ("pm2_18d" "ph" "th" ("syl6" "ph" ("wn" "th") ("wa" "ch" "ta") "th" ("jcad" "ph" ("wn" "th") "ch" "ta" ("syl5bir" ("wn" "th") ("wn" "ps") "ph" "ch" ("notbii" "ps" "th" "oplem1_3") ("ord" "ph" "ps" "ch" "oplem1_1")) ("ord" "ph" "th" "ta" "oplem1_2")) ("biimpar" "ch" "th" "ta" "oplem1_4"))) "oplem1_3")) ;; A single axiom for Boolean algebra known as DN_1. See McCune, Veroff, Fitelson, Harris, Feist, Wos, Short single axioms for Boolean algebra, Journal of Automated Reasoning, 29(1):1--16, 2002. ( ~ https://www.cs.unm.edu/~~mccune/papers/basax/v12.pdf ). (Contributed by Jeff Hankins, 3-Jul-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 6-Jan-2013.) (theorem "dn1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wn" ("wo" ("wn" ("wo" ("wn" ("wo" "ph" "ps")) "ch")) ("wn" ("wo" "ph" ("wn" ("wo" ("wn" "ch") ("wn" ("wo" "ch" "th")))))))) "ch") ("_3bitrri" "ch" ("wa" ("wo" ("wn" ("wo" "ph" "ps")) "ch") ("wo" "ph" "ch")) ("wa" ("wo" ("wn" ("wo" "ph" "ps")) "ch") ("wo" "ph" ("wn" ("wo" ("wn" "ch") ("wn" ("wo" "ch" "th")))))) ("wn" ("wo" ("wn" ("wo" ("wn" ("wo" "ph" "ps")) "ch")) ("wn" ("wo" "ph" ("wn" ("wo" ("wn" "ch") ("wn" ("wo" "ch" "th")))))))) ("bitri" "ch" ("wo" "ch" ("wa" ("wn" ("wo" "ph" "ps")) "ph")) ("wa" ("wo" ("wn" ("wo" "ph" "ps")) "ch") ("wo" "ph" "ch")) ("biorfi" ("wa" ("wn" ("wo" "ph" "ps")) "ph") "ch" ("mpbi" ("wi" ("wn" ("wo" "ph" "ps")) ("wn" "ph")) ("wn" ("wa" ("wn" ("wo" "ph" "ps")) "ph")) ("pm2_45" "ph" "ps") ("imnan" ("wn" ("wo" "ph" "ps")) "ph"))) ("bitri" ("wo" "ch" ("wa" ("wn" ("wo" "ph" "ps")) "ph")) ("wo" ("wa" ("wn" ("wo" "ph" "ps")) "ph") "ch") ("wa" ("wo" ("wn" ("wo" "ph" "ps")) "ch") ("wo" "ph" "ch")) ("orcom" "ch" ("wa" ("wn" ("wo" "ph" "ps")) "ph")) ("ordir" ("wn" ("wo" "ph" "ps")) "ph" "ch"))) ("anbi2i" ("wo" "ph" "ch") ("wo" "ph" ("wn" ("wo" ("wn" "ch") ("wn" ("wo" "ch" "th"))))) ("wo" ("wn" ("wo" "ph" "ps")) "ch") ("orbi2i" "ch" ("wn" ("wo" ("wn" "ch") ("wn" ("wo" "ch" "th")))) "ph" ("bitri" "ch" ("wa" "ch" ("wo" "ch" "th")) ("wn" ("wo" ("wn" "ch") ("wn" ("wo" "ch" "th")))) ("pm4_45" "ch" "th") ("anor" "ch" ("wo" "ch" "th"))))) ("anor" ("wo" ("wn" ("wo" "ph" "ps")) "ch") ("wo" "ph" ("wn" ("wo" ("wn" "ch") ("wn" ("wo" "ch" "th")))))))) ;; A closed form of ~ mpbir , analogous to ~ pm2.27 (assertion). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Roger Witte, 17-Aug-2020.) (theorem "bianir" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("wb" "ps" "ph")) "ps") ("impcom" ("wb" "ps" "ph") "ph" "ps" ("biimpr" "ps" "ph"))) ;; Inference removing a negated conjunct in a disjunction of an antecedent if this conjunct is part of the disjunction. (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (theorem "jaoi2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("jaoi2_1" ("wi" ("wo" "ph" ("wa" ("wn" "ph") "ch")) "ps"))) (for) ("wi" ("wo" "ph" "ch") "ps") ("sylbi" ("wo" "ph" "ch") ("wo" "ph" ("wa" ("wn" "ph") "ch")) "ps" ("pm5_63" "ph" "ch") "jaoi2_1")) ;; Inference separating a disjunct of an antecedent. (Contributed by Alexander van der Vekens, 25-May-2018.) (theorem "jaoi3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("jaoi3_1" ("wi" "ph" "ps")) ("jaoi3_2" ("wi" ("wa" ("wn" "ph") "ch") "ps"))) (for) ("wi" ("wo" "ph" "ch") "ps") ("jaoi2" "ph" "ps" "ch" ("jaoi" "ph" "ps" ("wa" ("wn" "ph") "ch") "jaoi3_1" "jaoi3_2"))) ;; Extend class notation to include the conditional operator for propositions. (term "wif" ( ( "wff") ( "wff") ( "wff") ( "wff"))) ;; Definition of the conditional operator for propositions. The value of ` if- ( ph , ps , ch ) ` is ` ps ` if ` ph ` is true and ` ch ` if ` ph ` false. See ~ dfifp2 , ~ dfifp3 , ~ dfifp4 , ~ dfifp5 , ~ dfifp6 and ~ dfifp7 for alternate definitions. This definition (in the form of ~ dfifp2 ) appears in Section II.24 of [Church] p. 129 (Definition D12 page 132), where it is called "conditioned disjunction". Church's ` [ ps , ph , ch ] ` corresponds to our ` if- ( ph , ps , ch ) ` (note the permutation of the first two variables). Church uses the conditional operator as an intermediate step to prove completeness of some systems of connectives. The first result is that the system ` { if- , T. , F. } ` is complete: for the induction step, consider a wff with n+1 variables; single out one variable, say ` ph ` ; when one sets ` ph ` to True (resp. False), then what remains is a wff of n variables, so by the induction hypothesis it corresponds to a formula using only ` { if- , T. , F. } ` , say ` ps ` (resp. ` ch ` ); therefore, the formula ` if- ( ph , ps , ch ) ` represents the initial wff. Now, since ` { -> , -. } ` and similar systems suffice to express ` if- , T. , F. ` , they are also complete. (Contributed by BJ, 22-Jun-2019.) (axiom "df_ifp" (!! ( ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) ("wb" ("wif" "ph" "ps" "ch") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch"))))) ;; Alternate definition of the conditional operator for propositions. The value of ` if- ( ph , ps , ch ) ` is "if ` ph ` then ` ps ` , and if not ` ph ` then ` ch ` ." This is the definition used in Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of ~ df-ifp ). (Contributed by BJ, 22-Jun-2019.) (theorem "dfifp2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wif" "ph" "ps" "ch") ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch"))) ("bitri" ("wif" "ph" "ps" "ch") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch")) ("df_ifp" "ph" "ps" "ch") ("cases2" "ph" "ps" "ch"))) ;; Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.) (theorem "dfifp3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wif" "ph" "ps" "ch") ("wa" ("wi" "ph" "ps") ("wo" "ph" "ch"))) ("bitri" ("wif" "ph" "ps" "ch") ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch")) ("wa" ("wi" "ph" "ps") ("wo" "ph" "ch")) ("dfifp2" "ph" "ps" "ch") ("anbi2i" ("wi" ("wn" "ph") "ch") ("wo" "ph" "ch") ("wi" "ph" "ps") ("pm4_64" "ph" "ch")))) ;; Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.) (theorem "dfifp4" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wif" "ph" "ps" "ch") ("wa" ("wo" ("wn" "ph") "ps") ("wo" "ph" "ch"))) ("bitri" ("wif" "ph" "ps" "ch") ("wa" ("wi" "ph" "ps") ("wo" "ph" "ch")) ("wa" ("wo" ("wn" "ph") "ps") ("wo" "ph" "ch")) ("dfifp3" "ph" "ps" "ch") ("anbi1i" ("wi" "ph" "ps") ("wo" ("wn" "ph") "ps") ("wo" "ph" "ch") ("imor" "ph" "ps")))) ;; Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.) (theorem "dfifp5" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wif" "ph" "ps" "ch") ("wa" ("wo" ("wn" "ph") "ps") ("wi" ("wn" "ph") "ch"))) ("bitri" ("wif" "ph" "ps" "ch") ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch")) ("wa" ("wo" ("wn" "ph") "ps") ("wi" ("wn" "ph") "ch")) ("dfifp2" "ph" "ps" "ch") ("anbi1i" ("wi" "ph" "ps") ("wo" ("wn" "ph") "ps") ("wi" ("wn" "ph") "ch") ("imor" "ph" "ps")))) ;; Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.) (theorem "dfifp6" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wif" "ph" "ps" "ch") ("wo" ("wa" "ph" "ps") ("wn" ("wi" "ch" "ph")))) ("bitri" ("wif" "ph" "ps" "ch") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wo" ("wa" "ph" "ps") ("wn" ("wi" "ch" "ph"))) ("df_ifp" "ph" "ps" "ch") ("orbi2i" ("wa" ("wn" "ph") "ch") ("wn" ("wi" "ch" "ph")) ("wa" "ph" "ps") ("bitri" ("wa" ("wn" "ph") "ch") ("wa" "ch" ("wn" "ph")) ("wn" ("wi" "ch" "ph")) ("ancom" ("wn" "ph") "ch") ("annim" "ch" "ph"))))) ;; Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.) (theorem "dfifp7" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wif" "ph" "ps" "ch") ("wi" ("wi" "ch" "ph") ("wa" "ph" "ps"))) ("_3bitr4i" ("wo" ("wa" "ph" "ps") ("wn" ("wi" "ch" "ph"))) ("wo" ("wn" ("wi" "ch" "ph")) ("wa" "ph" "ps")) ("wif" "ph" "ps" "ch") ("wi" ("wi" "ch" "ph") ("wa" "ph" "ps")) ("orcom" ("wa" "ph" "ps") ("wn" ("wi" "ch" "ph"))) ("dfifp6" "ph" "ps" "ch") ("imor" ("wi" "ch" "ph") ("wa" "ph" "ps")))) ;; The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ~ ifpor . (Contributed by BJ, 30-Sep-2019.) (theorem "anifp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" "ps" "ch") ("wif" "ph" "ps" "ch")) ("sylibr" ("wa" "ps" "ch") ("wa" ("wo" ("wn" "ph") "ps") ("wo" "ph" "ch")) ("wif" "ph" "ps" "ch") ("anim12i" "ps" ("wo" ("wn" "ph") "ps") "ch" ("wo" "ph" "ch") ("olc" "ps" ("wn" "ph")) ("olc" "ch" "ph")) ("dfifp4" "ph" "ps" "ch"))) ;; The conditional operator implies the disjunction of its possible outputs. Dual statement of ~ anifp . (Contributed by BJ, 1-Oct-2019.) (theorem "ifpor" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wif" "ph" "ps" "ch") ("wo" "ps" "ch")) ("sylbi" ("wif" "ph" "ps" "ch") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) ("wo" "ps" "ch") ("df_ifp" "ph" "ps" "ch") ("orim12i" ("wa" "ph" "ps") "ps" ("wa" ("wn" "ph") "ch") "ch" ("simpr" "ph" "ps") ("simpr" ("wn" "ph") "ch")))) ;; Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.) (theorem "ifpn" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wif" "ph" "ps" "ch") ("wif" ("wn" "ph") "ch" "ps")) ("_3bitr4i" ("wa" ("wi" "ph" "ps") ("wi" ("wn" "ph") "ch")) ("wa" ("wi" ("wn" "ph") "ch") ("wi" ("wn" ("wn" "ph")) "ps")) ("wif" "ph" "ps" "ch") ("wif" ("wn" "ph") "ch" "ps") ("anbi2ci" ("wi" "ph" "ps") ("wi" ("wn" ("wn" "ph")) "ps") ("wi" ("wn" "ph") "ch") ("imbi1i" "ph" ("wn" ("wn" "ph")) "ps" ("notnotb" "ph"))) ("dfifp2" "ph" "ps" "ch") ("dfifp2" ("wn" "ph") "ch" "ps"))) ;; Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of ~ iftrue . This is essentially ~ dedlema . (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.) (theorem "ifptru" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" "ph" ("wb" ("wif" "ph" "ps" "ch") "ps")) ("bitr2d" "ph" "ps" ("wi" "ph" "ps") ("wif" "ph" "ps" "ch") ("biimt" "ph" "ps") ("syl6bbr" "ph" ("wi" "ph" "ps") ("wa" ("wi" "ph" "ps") ("wo" "ph" "ch")) ("wif" "ph" "ps" "ch") ("biantrud" "ph" ("wo" "ph" "ch") ("wi" "ph" "ps") ("orc" "ph" "ch")) ("dfifp3" "ph" "ps" "ch")))) ;; Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of ~ iffalse . This is essentially ~ dedlemb . (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.) (theorem "ifpfal" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wb" ("wif" "ph" "ps" "ch") "ch")) ("syl5bb" ("wif" "ph" "ps" "ch") ("wif" ("wn" "ph") "ch" "ps") ("wn" "ph") "ch" ("ifpn" "ph" "ps" "ch") ("ifptru" ("wn" "ph") "ch" "ps"))) ;; Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ~ ifid . This is essentially ~ pm4.42 . (Contributed by BJ, 20-Sep-2019.) (theorem "ifpid" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wif" "ph" "ps" "ps") "ps") ("pm2_61i" "ph" ("wb" ("wif" "ph" "ps" "ps") "ps") ("ifptru" "ph" "ps" "ps") ("ifpfal" "ph" "ps" "ps"))) ;; Version of ~ cases expressed using ` if- ` . Case disjunction according to the value of ` ph ` . One can see this as a proof that the two hypotheses characterize the conditional operator for propositions. For the converses, see ~ ifptru and ~ ifpfal . (Contributed by BJ, 20-Sep-2019.) (theorem "casesifp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("casesifp_1" ("wi" "ph" ("wb" "ps" "ch"))) ("casesifp_2" ("wi" ("wn" "ph") ("wb" "ps" "th")))) (for) ("wb" "ps" ("wif" "ph" "ch" "th")) ("bitr4i" "ps" ("wo" ("wa" "ph" "ch") ("wa" ("wn" "ph") "th")) ("wif" "ph" "ch" "th") ("cases" "ph" "ps" "ch" "th" "casesifp_1" "casesifp_2") ("df_ifp" "ph" "ch" "th"))) ;; Equality deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (theorem "ifpbi123d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("ifpbi123d_1" ("wi" "ph" ("wb" "ps" "ta"))) ("ifpbi123d_2" ("wi" "ph" ("wb" "ch" "et"))) ("ifpbi123d_3" ("wi" "ph" ("wb" "th" "ze")))) (for) ("wi" "ph" ("wb" ("wif" "ps" "ch" "th") ("wif" "ta" "et" "ze"))) ("_3bitr4g" "ph" ("wo" ("wa" "ps" "ch") ("wa" ("wn" "ps") "th")) ("wo" ("wa" "ta" "et") ("wa" ("wn" "ta") "ze")) ("wif" "ps" "ch" "th") ("wif" "ta" "et" "ze") ("orbi12d" "ph" ("wa" "ps" "ch") ("wa" "ta" "et") ("wa" ("wn" "ps") "th") ("wa" ("wn" "ta") "ze") ("anbi12d" "ph" "ps" "ta" "ch" "et" "ifpbi123d_1" "ifpbi123d_2") ("anbi12d" "ph" ("wn" "ps") ("wn" "ta") "th" "ze" ("notbid" "ph" "ps" "ta" "ifpbi123d_1") "ifpbi123d_3")) ("df_ifp" "ps" "ch" "th") ("df_ifp" "ta" "et" "ze"))) ;; Separation of the values of the conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 27-Feb-2021.) (theorem "ifpimpda" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("ifpimpda_1" ("wi" ("wa" "ph" "ps") "ch")) ("ifpimpda_2" ("wi" ("wa" "ph" ("wn" "ps")) "th"))) (for) ("wi" "ph" ("wif" "ps" "ch" "th")) ("sylanbrc" "ph" ("wi" "ps" "ch") ("wi" ("wn" "ps") "th") ("wif" "ps" "ch" "th") ("ex" "ph" "ps" "ch" "ifpimpda_1") ("ex" "ph" ("wn" "ps") "th" "ifpimpda_2") ("dfifp2" "ps" "ch" "th"))) ;; The value of the conditional operator for propositions is its third argument if the first and second argument imply the third argument. (Contributed by AV, 4-Apr-2021.) (theorem "_1fpid3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_1fpid3_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wif" "ph" "ps" "ch") "ch") ("sylbi" ("wif" "ph" "ps" "ch") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") "ch")) "ch" ("df_ifp" "ph" "ps" "ch") ("jaoi" ("wa" "ph" "ps") "ch" ("wa" ("wn" "ph") "ch") "_1fpid3_1" ("simpr" ("wn" "ph") "ch")))) ;; Hypothesis builder for the weak deduction theorem. For more information, see the Weak Deduction Theorem page ~ mmdeduction.html . (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) (theorem "elimh" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("elimh_1" ("wi" ("wb" ("wif" "ch" "ph" "ps") "ph") ("wb" "ch" "ta"))) ("elimh_2" ("wi" ("wb" ("wif" "ch" "ph" "ps") "ps") ("wb" "th" "ta"))) ("elimh_3" "th")) (for) "ta" ("pm2_61i" "ch" "ta" ("ibi" "ch" "ta" ("syl" "ch" ("wb" ("wif" "ch" "ph" "ps") "ph") ("wb" "ch" "ta") ("ifptru" "ch" "ph" "ps") "elimh_1")) ("mpbii" ("wn" "ch") "th" "ta" "elimh_3" ("syl" ("wn" "ch") ("wb" ("wif" "ch" "ph" "ps") "ps") ("wb" "th" "ta") ("ifpfal" "ch" "ph" "ps") "elimh_2")))) ;; The weak deduction theorem. For more information, see the Weak Deduction Theorem page ~ mmdeduction.html . (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) (theorem "dedt" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("dedt_1" ("wi" ("wb" ("wif" "ch" "ph" "ps") "ph") ("wb" "th" "ta"))) ("dedt_2" "ta")) (for) ("wi" "ch" "th") ("syl" "ch" ("wb" ("wif" "ch" "ph" "ps") "ph") "th" ("ifptru" "ch" "ph" "ps") ("mpbiri" ("wb" ("wif" "ch" "ph" "ps") "ph") "th" "ta" "dedt_2" "dedt_1"))) ;; Proof of ~ con3 from its associated inference ~ con3i that illustrates the use of the weak deduction theorem ~ dedt . (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "con3ALT" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wn" "ps") ("wn" "ph"))) ("dedt" "ps" "ph" ("wi" "ph" "ps") ("wi" ("wn" "ps") ("wn" "ph")) ("wi" ("wn" ("wif" ("wi" "ph" "ps") "ps" "ph")) ("wn" "ph")) ("imbi1d" ("wb" ("wif" ("wi" "ph" "ps") "ps" "ph") "ps") ("wn" "ps") ("wn" ("wif" ("wi" "ph" "ps") "ps" "ph")) ("wn" "ph") ("notbid" ("wb" ("wif" ("wi" "ph" "ps") "ps" "ph") "ps") "ps" ("wif" ("wi" "ph" "ps") "ps" "ph") ("bicom1" ("wif" ("wi" "ph" "ps") "ps" "ph") "ps"))) ("con3i" "ph" ("wif" ("wi" "ph" "ps") "ps" "ph") ("elimh" "ps" "ph" ("wi" "ph" "ps") ("wi" "ph" "ph") ("wi" "ph" ("wif" ("wi" "ph" "ps") "ps" "ph")) ("imbi2d" ("wb" ("wif" ("wi" "ph" "ps") "ps" "ph") "ps") "ps" ("wif" ("wi" "ph" "ps") "ps" "ph") "ph" ("bicom1" ("wif" ("wi" "ph" "ps") "ps" "ph") "ps")) ("imbi2d" ("wb" ("wif" ("wi" "ph" "ps") "ps" "ph") "ph") "ph" ("wif" ("wi" "ph" "ps") "ps" "ph") "ph" ("bicom1" ("wif" ("wi" "ph" "ps") "ps" "ph") "ph")) ("id" "ph"))))) ;; Extend wff definition to include three-way disjunction ('or'). (term "w3o" ( ( "wff") ( "wff") ( "wff") ( "wff"))) ;; Extend wff definition to include three-way conjunction ('and'). (term "w3a" ( ( "wff") ( "wff") ( "wff") ( "wff"))) ;; Define disjunction ('or') of three wff's. Definition *2.33 of [WhiteheadRussell] p. 105. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law ~ orass . (Contributed by NM, 8-Apr-1994.) (axiom "df_3or" (!! ( ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) ("wb" ("w3o" "ph" "ps" "ch") ("wo" ("wo" "ph" "ps") "ch")))) ;; Define conjunction ('and') of three wff's. Definition *4.34 of [WhiteheadRussell] p. 118. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law ~ anass . (Contributed by NM, 8-Apr-1994.) (axiom "df_3an" (!! ( ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) ("wb" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") "ch")))) ;; Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.) (theorem "_3orass" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3o" "ph" "ps" "ch") ("wo" "ph" ("wo" "ps" "ch"))) ("bitri" ("w3o" "ph" "ps" "ch") ("wo" ("wo" "ph" "ps") "ch") ("wo" "ph" ("wo" "ps" "ch")) ("df_3or" "ph" "ps" "ch") ("orass" "ph" "ps" "ch"))) ;; Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) (theorem "_3orel1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wi" ("w3o" "ph" "ps" "ch") ("wo" "ps" "ch"))) ("syl5bi" ("w3o" "ph" "ps" "ch") ("wo" "ph" ("wo" "ps" "ch")) ("wn" "ph") ("wo" "ps" "ch") ("_3orass" "ph" "ps" "ch") ("orel1" "ph" ("wo" "ps" "ch")))) ;; Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.) (theorem "_3orrot" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3o" "ph" "ps" "ch") ("w3o" "ps" "ch" "ph")) ("_3bitr4i" ("wo" "ph" ("wo" "ps" "ch")) ("wo" ("wo" "ps" "ch") "ph") ("w3o" "ph" "ps" "ch") ("w3o" "ps" "ch" "ph") ("orcom" "ph" ("wo" "ps" "ch")) ("_3orass" "ph" "ps" "ch") ("df_3or" "ps" "ch" "ph"))) ;; Commutation law for triple disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "_3orcoma" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3o" "ph" "ps" "ch") ("w3o" "ps" "ph" "ch")) ("_3bitr4i" ("wo" "ph" ("wo" "ps" "ch")) ("wo" "ps" ("wo" "ph" "ch")) ("w3o" "ph" "ps" "ch") ("w3o" "ps" "ph" "ch") ("or12" "ph" "ps" "ch") ("_3orass" "ph" "ps" "ch") ("_3orass" "ps" "ph" "ch"))) ;; Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.) (Proof shortened by Wolf Lammen, 8-Apr-2022.) (theorem "_3orcomb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3o" "ph" "ps" "ch") ("w3o" "ph" "ch" "ps")) ("bitri" ("w3o" "ph" "ps" "ch") ("w3o" "ps" "ph" "ch") ("w3o" "ph" "ch" "ps") ("_3orcoma" "ph" "ps" "ch") ("_3orrot" "ps" "ph" "ch"))) ;; Obsolete version of ~ 3orcomb as of 8-Apr-2022. (Contributed by Scott Fenton, 20-Apr-2011.) (New usage is discouraged.) (Proof modification is discouraged.) (theorem "_3orcombOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3o" "ph" "ps" "ch") ("w3o" "ph" "ch" "ps")) ("_3bitr4i" ("wo" "ph" ("wo" "ps" "ch")) ("wo" "ph" ("wo" "ch" "ps")) ("w3o" "ph" "ps" "ch") ("w3o" "ph" "ch" "ps") ("orbi2i" ("wo" "ps" "ch") ("wo" "ch" "ps") "ph" ("orcom" "ps" "ch")) ("_3orass" "ph" "ps" "ch") ("_3orass" "ph" "ch" "ps"))) ;; Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.) (theorem "_3anass" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("wa" "ph" ("wa" "ps" "ch"))) ("bitri" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") "ch") ("wa" "ph" ("wa" "ps" "ch")) ("df_3an" "ph" "ps" "ch") ("anass" "ph" "ps" "ch"))) ;; Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised to shorten ~ 3ancoma by Wolf Lammen, 5-Jun-2022.) (theorem "_3anan12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("wa" "ps" ("wa" "ph" "ch"))) ("bitri" ("w3a" "ph" "ps" "ch") ("wa" "ph" ("wa" "ps" "ch")) ("wa" "ps" ("wa" "ph" "ch")) ("_3anass" "ph" "ps" "ch") ("an12" "ph" "ps" "ch"))) ;; Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (theorem "_3anan32" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ch") "ps")) ("bitri" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") "ch") ("wa" ("wa" "ph" "ch") "ps") ("df_3an" "ph" "ps" "ch") ("an32" "ph" "ps" "ch"))) ;; Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 5-Jun-2022.) (theorem "_3ancoma" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("w3a" "ps" "ph" "ch")) ("bitr4i" ("w3a" "ph" "ps" "ch") ("wa" "ps" ("wa" "ph" "ch")) ("w3a" "ps" "ph" "ch") ("_3anan12" "ph" "ps" "ch") ("_3anass" "ps" "ph" "ch"))) ;; Obsolete version of ~ 3ancoma as of 5-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3ancomaOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("w3a" "ps" "ph" "ch")) ("_3bitr4i" ("wa" ("wa" "ph" "ps") "ch") ("wa" ("wa" "ps" "ph") "ch") ("w3a" "ph" "ps" "ch") ("w3a" "ps" "ph" "ch") ("anbi1i" ("wa" "ph" "ps") ("wa" "ps" "ph") "ch" ("ancom" "ph" "ps")) ("df_3an" "ph" "ps" "ch") ("df_3an" "ps" "ph" "ch"))) ;; Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.) (Revised to shorten ~ 3anrot by Wolf Lammen, 9-Jun-2022.) (theorem "_3ancomb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("w3a" "ph" "ch" "ps")) ("bitr4i" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") "ch") ("w3a" "ph" "ch" "ps") ("df_3an" "ph" "ps" "ch") ("_3anan32" "ph" "ch" "ps"))) ;; Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.) (Proof shortened by Wolf Lammen, 9-Jun-2022.) (theorem "_3anrot" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("w3a" "ps" "ch" "ph")) ("bitri" ("w3a" "ph" "ps" "ch") ("w3a" "ps" "ph" "ch") ("w3a" "ps" "ch" "ph") ("_3ancoma" "ph" "ps" "ch") ("_3ancomb" "ps" "ph" "ch"))) ;; Obsolete version of ~ 3anan12 as of 5-Jun-2022. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3anan12OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("wa" "ps" ("wa" "ph" "ch"))) ("bitri" ("w3a" "ph" "ps" "ch") ("w3a" "ps" "ph" "ch") ("wa" "ps" ("wa" "ph" "ch")) ("_3ancoma" "ph" "ps" "ch") ("_3anass" "ps" "ph" "ch"))) ;; Obsolete version of ~ 3anrot as of 9-Jun-2022. (Contributed by NM, 8-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3anrotOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("w3a" "ps" "ch" "ph")) ("_3bitr4i" ("wa" "ph" ("wa" "ps" "ch")) ("wa" ("wa" "ps" "ch") "ph") ("w3a" "ph" "ps" "ch") ("w3a" "ps" "ch" "ph") ("ancom" "ph" ("wa" "ps" "ch")) ("_3anass" "ph" "ps" "ch") ("df_3an" "ps" "ch" "ph"))) ;; Obsolete version of ~ 3ancomb as of 9-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3ancombOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("w3a" "ph" "ch" "ps")) ("bitri" ("w3a" "ph" "ps" "ch") ("w3a" "ps" "ph" "ch") ("w3a" "ph" "ch" "ps") ("_3ancoma" "ph" "ps" "ch") ("_3anrot" "ps" "ph" "ch"))) ;; Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.) (theorem "_3anrev" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("w3a" "ch" "ps" "ph")) ("bitr4i" ("w3a" "ph" "ps" "ch") ("w3a" "ps" "ph" "ch") ("w3a" "ch" "ps" "ph") ("_3ancoma" "ph" "ps" "ch") ("_3anrot" "ch" "ps" "ph"))) ;; Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.) (theorem "anandi3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") ("wa" "ph" "ch"))) ("bitri" ("w3a" "ph" "ps" "ch") ("wa" "ph" ("wa" "ps" "ch")) ("wa" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("_3anass" "ph" "ps" "ch") ("anandi" "ph" "ps" "ch"))) ;; Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.) (theorem "anandi3r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") ("wa" "ch" "ps"))) ("bitri" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ch") "ps") ("wa" ("wa" "ph" "ps") ("wa" "ch" "ps")) ("_3anan32" "ph" "ps" "ch") ("anandir" "ph" "ch" "ps"))) ;; Obsolete version of ~ 3anor as of 8-Apr-2022. (Contributed by Jeff Hankins, 15-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.) (theorem "_3anorOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("wn" ("w3o" ("wn" "ph") ("wn" "ps") ("wn" "ch")))) ("bitri" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") "ch") ("wn" ("w3o" ("wn" "ph") ("wn" "ps") ("wn" "ch"))) ("df_3an" "ph" "ps" "ch") ("xchbinxr" ("wa" ("wa" "ph" "ps") "ch") ("wo" ("wo" ("wn" "ph") ("wn" "ps")) ("wn" "ch")) ("w3o" ("wn" "ph") ("wn" "ps") ("wn" "ch")) ("xchbinx" ("wa" ("wa" "ph" "ps") "ch") ("wo" ("wn" ("wa" "ph" "ps")) ("wn" "ch")) ("wo" ("wo" ("wn" "ph") ("wn" "ps")) ("wn" "ch")) ("anor" ("wa" "ph" "ps") "ch") ("orbi1i" ("wn" ("wa" "ph" "ps")) ("wo" ("wn" "ph") ("wn" "ps")) ("wn" "ch") ("ianor" "ph" "ps"))) ("df_3or" ("wn" "ph") ("wn" "ps") ("wn" "ch"))))) ;; Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.) (theorem "_3ioran" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wn" ("w3o" "ph" "ps" "ch")) ("w3a" ("wn" "ph") ("wn" "ps") ("wn" "ch"))) ("_3bitr4i" ("wa" ("wn" ("wo" "ph" "ps")) ("wn" "ch")) ("wa" ("wa" ("wn" "ph") ("wn" "ps")) ("wn" "ch")) ("wn" ("w3o" "ph" "ps" "ch")) ("w3a" ("wn" "ph") ("wn" "ps") ("wn" "ch")) ("anbi1i" ("wn" ("wo" "ph" "ps")) ("wa" ("wn" "ph") ("wn" "ps")) ("wn" "ch") ("ioran" "ph" "ps")) ("xchnxbir" ("wo" ("wo" "ph" "ps") "ch") ("wa" ("wn" ("wo" "ph" "ps")) ("wn" "ch")) ("w3o" "ph" "ps" "ch") ("ioran" ("wo" "ph" "ps") "ch") ("df_3or" "ph" "ps" "ch")) ("df_3an" ("wn" "ph") ("wn" "ps") ("wn" "ch")))) ;; Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Revised by Wolf Lammen, 8-Apr-2022.) (theorem "_3ianor" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wn" ("w3a" "ph" "ps" "ch")) ("w3o" ("wn" "ph") ("wn" "ps") ("wn" "ch"))) ("_3bitr4i" ("wo" ("wn" ("wa" "ph" "ps")) ("wn" "ch")) ("wo" ("wo" ("wn" "ph") ("wn" "ps")) ("wn" "ch")) ("wn" ("w3a" "ph" "ps" "ch")) ("w3o" ("wn" "ph") ("wn" "ps") ("wn" "ch")) ("orbi1i" ("wn" ("wa" "ph" "ps")) ("wo" ("wn" "ph") ("wn" "ps")) ("wn" "ch") ("ianor" "ph" "ps")) ("xchnxbir" ("wa" ("wa" "ph" "ps") "ch") ("wo" ("wn" ("wa" "ph" "ps")) ("wn" "ch")) ("w3a" "ph" "ps" "ch") ("ianor" ("wa" "ph" "ps") "ch") ("df_3an" "ph" "ps" "ch")) ("df_3or" ("wn" "ph") ("wn" "ps") ("wn" "ch")))) ;; Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Wolf Lammen, 8-Apr-2022.) (theorem "_3anor" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3a" "ph" "ps" "ch") ("wn" ("w3o" ("wn" "ph") ("wn" "ps") ("wn" "ch")))) ("bicomi" ("wn" ("w3o" ("wn" "ph") ("wn" "ps") ("wn" "ch"))) ("w3a" "ph" "ps" "ch") ("con1bii" ("w3a" "ph" "ps" "ch") ("w3o" ("wn" "ph") ("wn" "ps") ("wn" "ch")) ("_3ianor" "ph" "ps" "ch")))) ;; Obsolete version of ~ 3ianor as of 8-Apr-2022. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3ianorOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wn" ("w3a" "ph" "ps" "ch")) ("w3o" ("wn" "ph") ("wn" "ps") ("wn" "ch"))) ("bicomi" ("w3o" ("wn" "ph") ("wn" "ps") ("wn" "ch")) ("wn" ("w3a" "ph" "ps" "ch")) ("con2bii" ("w3a" "ph" "ps" "ch") ("w3o" ("wn" "ph") ("wn" "ps") ("wn" "ch")) ("_3anor" "ph" "ps" "ch")))) ;; Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012.) (theorem "_3oran" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("w3o" "ph" "ps" "ch") ("wn" ("w3a" ("wn" "ph") ("wn" "ps") ("wn" "ch")))) ("bicomi" ("wn" ("w3a" ("wn" "ph") ("wn" "ps") ("wn" "ch"))) ("w3o" "ph" "ps" "ch") ("con1bii" ("w3o" "ph" "ps" "ch") ("w3a" ("wn" "ph") ("wn" "ps") ("wn" "ch")) ("_3ioran" "ph" "ps" "ch")))) ;; Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten ~ 3imp by Wolf Lammen, 20-Jun-2022.) (theorem "_3impa" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3impa_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("w3a" "ph" "ps" "ch") "th") ("sylbi" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") "ch") "th" ("df_3an" "ph" "ps" "ch") "_3impa_1")) ;; Importation inference. (Contributed by NM, 8-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jun-2022.) (theorem "_3imp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3imp_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" ("w3a" "ph" "ps" "ch") "th") ("_3impa" "ph" "ps" "ch" "th" ("imp31" "ph" "ps" "ch" "th" "_3imp_1"))) ;; Obsolete version of ~ 3imp as of 20-Jun-2022. (Contributed by NM, 8-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3impOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3imp_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" ("w3a" "ph" "ps" "ch") "th") ("sylbi" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") "ch") "th" ("df_3an" "ph" "ps" "ch") ("imp31" "ph" "ps" "ch" "th" "_3imp_1"))) ;; The importation inference ~ 3imp with commutation of the first and third conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.) (theorem "_3imp31" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3imp_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" ("w3a" "ch" "ps" "ph") "th") ("_3imp" "ch" "ps" "ph" "th" ("com13" "ph" "ps" "ch" "th" "_3imp_1"))) ;; Importation inference. (Contributed by Alan Sare, 17-Oct-2017.) (theorem "_3imp231" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3imp_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" ("w3a" "ps" "ch" "ph") "th") ("_3imp" "ps" "ch" "ph" "th" ("com3l" "ph" "ps" "ch" "th" "_3imp_1"))) ;; The importation inference ~ 3imp with commutation of the first and second conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.) (Revised to shorten ~ 3com12 by Wolf Lammen, 23-Jun-2022.) (theorem "_3imp21" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3imp_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" ("w3a" "ps" "ph" "ch") "th") ("_3imp231" "ch" "ps" "ph" "th" ("com13" "ph" "ps" "ch" "th" "_3imp_1"))) ;; Obsolete version of ~ 3impa as of 20-Jun-2022. (Contributed by NM, 20-Aug-1995.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3impaOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3impaOLD_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("w3a" "ph" "ps" "ch") "th") ("_3imp" "ph" "ps" "ch" "th" ("exp31" "ph" "ps" "ch" "th" "_3impaOLD_1"))) ;; Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.) (theorem "_3impb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3impb_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("w3a" "ph" "ps" "ch") "th") ("_3imp" "ph" "ps" "ch" "th" ("exp32" "ph" "ps" "ch" "th" "_3impb_1"))) ;; Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) (theorem "_3impib" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3impib_1" ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")))) (for) ("wi" ("w3a" "ph" "ps" "ch") "th") ("_3imp" "ph" "ps" "ch" "th" ("expd" "ph" "ps" "ch" "th" "_3impib_1"))) ;; Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) (theorem "_3impia" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3impia_1" ("wi" ("wa" "ph" "ps") ("wi" "ch" "th")))) (for) ("wi" ("w3a" "ph" "ps" "ch") "th") ("_3impib" "ph" "ps" "ch" "th" ("expimpd" "ph" "ps" "ch" "th" "_3impia_1"))) ;; Obsolete version of ~ 3impia as of 21-Jun-2022. (Contributed by NM, 13-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3impiaOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3impiaOLD_1" ("wi" ("wa" "ph" "ps") ("wi" "ch" "th")))) (for) ("wi" ("w3a" "ph" "ps" "ch") "th") ("_3imp" "ph" "ps" "ch" "th" ("ex" "ph" "ps" ("wi" "ch" "th") "_3impiaOLD_1"))) ;; Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten ~ 3exp and ~ pm3.2an3 by Wolf Lammen, 22-Jun-2022.) (theorem "_3expa" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3exp_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" "ph" "ps") "ch") "th") ("sylbir" ("wa" ("wa" "ph" "ps") "ch") ("w3a" "ph" "ps" "ch") "th" ("df_3an" "ph" "ps" "ch") "_3exp_1")) ;; Exportation inference. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) (theorem "_3exp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3exp_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))) ("exp31" "ph" "ps" "ch" "th" ("_3expa" "ph" "ps" "ch" "th" "_3exp_1"))) ;; Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.) (theorem "_3expb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3exp_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th") ("imp32" "ph" "ps" "ch" "th" ("_3exp" "ph" "ps" "ch" "th" "_3exp_1"))) ;; Exportation from triple conjunction. (Contributed by NM, 19-May-2007.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) (theorem "_3expia" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3exp_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" "ps") ("wi" "ch" "th")) ("expr" "ph" "ps" "ch" "th" ("_3expb" "ph" "ps" "ch" "th" "_3exp_1"))) ;; Obsolete version of ~ 3expia as of 22-Jun-2022. (Contributed by NM, 19-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3expiaOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3exp_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" "ps") ("wi" "ch" "th")) ("imp" "ph" "ps" ("wi" "ch" "th") ("_3exp" "ph" "ps" "ch" "th" "_3exp_1"))) ;; Exportation from triple conjunction. (Contributed by NM, 19-May-2007.) (theorem "_3expib" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3exp_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" "ph" ("wi" ("wa" "ps" "ch") "th")) ("impd" "ph" "ps" "ch" "th" ("_3exp" "ph" "ps" "ch" "th" "_3exp_1"))) ;; Commutation in antecedent. Swap 1st and 2nd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) (theorem "_3com12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3exp_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ps" "ph" "ch") "th") ("_3imp21" "ph" "ps" "ch" "th" ("_3exp" "ph" "ps" "ch" "th" "_3exp_1"))) ;; Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) (theorem "_3com13" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3exp_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ch" "ps" "ph") "th") ("_3imp31" "ph" "ps" "ch" "th" ("_3exp" "ph" "ps" "ch" "th" "_3exp_1"))) ;; Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.) (Revised by Wolf Lammen, 9-Apr-2022.) (theorem "_3comr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3exp_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ch" "ph" "ps") "th") ("_3com13" "ps" "ph" "ch" "th" ("_3com12" "ph" "ps" "ch" "th" "_3exp_1"))) ;; Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Wolf Lammen, 9-Apr-2022.) (theorem "_3com23" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3exp_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ph" "ch" "ps") "th") ("_3com12" "ch" "ph" "ps" "th" ("_3comr" "ph" "ps" "ch" "th" "_3exp_1"))) ;; Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.) (theorem "_3coml" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3exp_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ps" "ch" "ph") "th") ("_3com13" "ph" "ch" "ps" "th" ("_3com23" "ph" "ps" "ch" "th" "_3exp_1"))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) (theorem "_3adant1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3adant_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("w3a" "th" "ph" "ps") "ch") ("_3impa" "th" "ph" "ps" "ch" ("adantll" "ph" "ps" "ch" "th" "_3adant_1"))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.) (theorem "_3adant2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3adant_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("w3a" "ph" "th" "ps") "ch") ("_3impa" "ph" "th" "ps" "ch" ("adantlr" "ph" "ps" "ch" "th" "_3adant_1"))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) (theorem "_3adant3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3adant_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("w3a" "ph" "ps" "th") "ch") ("_3impb" "ph" "ps" "th" "ch" ("adantrr" "ph" "ps" "ch" "th" "_3adant_1"))) ;; Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.) (theorem "_3ad2ant1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3ad2ant_1" ("wi" "ph" "ch"))) (for) ("wi" ("w3a" "ph" "ps" "th") "ch") ("_3adant2" "ph" "th" "ch" "ps" ("adantr" "ph" "ch" "th" "_3ad2ant_1"))) ;; Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.) (theorem "_3ad2ant2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3ad2ant_1" ("wi" "ph" "ch"))) (for) ("wi" ("w3a" "ps" "ph" "th") "ch") ("_3adant1" "ph" "th" "ch" "ps" ("adantr" "ph" "ch" "th" "_3ad2ant_1"))) ;; Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.) (theorem "_3ad2ant3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3ad2ant_1" ("wi" "ph" "ch"))) (for) ("wi" ("w3a" "ps" "th" "ph") "ch") ("_3adant1" "th" "ph" "ch" "ps" ("adantl" "ph" "ch" "th" "_3ad2ant_1"))) ;; Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) (theorem "simp1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") "ph") ("_3ad2ant1" "ph" "ps" "ph" "ch" ("id" "ph"))) ;; Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) (theorem "simp2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") "ps") ("_3ad2ant2" "ps" "ph" "ps" "ch" ("id" "ps"))) ;; Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) (theorem "simp3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") "ch") ("_3ad2ant3" "ch" "ph" "ch" "ps" ("id" "ch"))) ;; Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.) (theorem "simp1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_3simp1i_1" ("w3a" "ph" "ps" "ch"))) (for) "ph" ("ax_mp" ("w3a" "ph" "ps" "ch") "ph" "_3simp1i_1" ("simp1" "ph" "ps" "ch"))) ;; Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.) (theorem "simp2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_3simp1i_1" ("w3a" "ph" "ps" "ch"))) (for) "ps" ("ax_mp" ("w3a" "ph" "ps" "ch") "ps" "_3simp1i_1" ("simp2" "ph" "ps" "ch"))) ;; Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.) (theorem "simp3i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_3simp1i_1" ("w3a" "ph" "ps" "ch"))) (for) "ch" ("ax_mp" ("w3a" "ph" "ps" "ch") "ch" "_3simp1i_1" ("simp3" "ph" "ps" "ch"))) ;; Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.) (theorem "simp1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3simp1d_1" ("wi" "ph" ("w3a" "ps" "ch" "th")))) (for) ("wi" "ph" "ps") ("syl" "ph" ("w3a" "ps" "ch" "th") "ps" "_3simp1d_1" ("simp1" "ps" "ch" "th"))) ;; Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.) (theorem "simp2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3simp1d_1" ("wi" "ph" ("w3a" "ps" "ch" "th")))) (for) ("wi" "ph" "ch") ("syl" "ph" ("w3a" "ps" "ch" "th") "ch" "_3simp1d_1" ("simp2" "ps" "ch" "th"))) ;; Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.) (theorem "simp3d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3simp1d_1" ("wi" "ph" ("w3a" "ps" "ch" "th")))) (for) ("wi" "ph" "th") ("syl" "ph" ("w3a" "ps" "ch" "th") "th" "_3simp1d_1" ("simp3" "ps" "ch" "th"))) ;; Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (theorem "simp1bi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3simp1bi_1" ("wb" "ph" ("w3a" "ps" "ch" "th")))) (for) ("wi" "ph" "ps") ("simp1d" "ph" "ps" "ch" "th" ("biimpi" "ph" ("w3a" "ps" "ch" "th") "_3simp1bi_1"))) ;; Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (theorem "simp2bi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3simp1bi_1" ("wb" "ph" ("w3a" "ps" "ch" "th")))) (for) ("wi" "ph" "ch") ("simp2d" "ph" "ps" "ch" "th" ("biimpi" "ph" ("w3a" "ps" "ch" "th") "_3simp1bi_1"))) ;; Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (theorem "simp3bi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3simp1bi_1" ("wb" "ph" ("w3a" "ps" "ch" "th")))) (for) ("wi" "ph" "th") ("simp3d" "ph" "ps" "ch" "th" ("biimpi" "ph" ("w3a" "ps" "ch" "th") "_3simp1bi_1"))) ;; Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) (theorem "_3simpa" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") ("wa" "ph" "ps")) ("_3adant3" "ph" "ps" ("wa" "ph" "ps") "ch" ("id" ("wa" "ph" "ps")))) ;; Obsolete version of ~ 3simpa as of 21-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3simpaOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") ("wa" "ph" "ps")) ("simplbi" ("w3a" "ph" "ps" "ch") ("wa" "ph" "ps") "ch" ("df_3an" "ph" "ps" "ch"))) ;; Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) (theorem "_3simpb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") ("wa" "ph" "ch")) ("_3adant2" "ph" "ch" ("wa" "ph" "ch") "ps" ("id" ("wa" "ph" "ch")))) ;; Obsolete version of ~ 3simpb as of 21-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3simpbOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") ("wa" "ph" "ch")) ("sylbi" ("w3a" "ph" "ps" "ch") ("w3a" "ph" "ch" "ps") ("wa" "ph" "ch") ("_3ancomb" "ph" "ps" "ch") ("_3simpa" "ph" "ch" "ps"))) ;; Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) (theorem "_3simpc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") ("wa" "ps" "ch")) ("_3adant1" "ps" "ch" ("wa" "ps" "ch") "ph" ("id" ("wa" "ps" "ch")))) ;; Obsolete version of ~ 3simpc as of 21-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3simpcOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") ("wa" "ps" "ch")) ("sylbi" ("w3a" "ph" "ps" "ch") ("w3a" "ps" "ch" "ph") ("wa" "ps" "ch") ("_3anrot" "ph" "ps" "ch") ("_3simpa" "ps" "ch" "ph"))) ;; Obsolete version of ~ simp1 as of 22-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp1OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") "ph") ("simpld" ("w3a" "ph" "ps" "ch") "ph" "ps" ("_3simpa" "ph" "ps" "ch"))) ;; Obsolete version of ~ simp2 as of 22-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp2OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") "ps") ("simprd" ("w3a" "ph" "ps" "ch") "ph" "ps" ("_3simpa" "ph" "ps" "ch"))) ;; Obsolete version of ~ simp3 as of 22-Jun-2022. (Contributed by NM, 21-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simp3OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" "ch") "ch") ("simprd" ("w3a" "ph" "ps" "ch") "ps" "ch" ("_3simpc" "ph" "ps" "ch"))) ;; Obsolete version of ~ 3adant1 as of 21-Jun-2022. (Contributed by NM, 16-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3adant1OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3adantOLD_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("w3a" "th" "ph" "ps") "ch") ("syl" ("w3a" "th" "ph" "ps") ("wa" "ph" "ps") "ch" ("_3simpc" "th" "ph" "ps") "_3adantOLD_1")) ;; Obsolete version of ~ 3adant1 as of 21-Jun-2022. (Contributed by NM, 16-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3adant2OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3adantOLD_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("w3a" "ph" "th" "ps") "ch") ("syl" ("w3a" "ph" "th" "ps") ("wa" "ph" "ps") "ch" ("_3simpb" "ph" "th" "ps") "_3adantOLD_1")) ;; Obsolete version of ~ 3adant3 as of 21-Jun-2022. (Contributed by NM, 16-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3adant3OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3adantOLD_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("w3a" "ph" "ps" "th") "ch") ("syl" ("w3a" "ph" "ps" "th") ("wa" "ph" "ps") "ch" ("_3simpa" "ph" "ps" "th") "_3adantOLD_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.) (theorem "_3adantl1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3adantl_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" ("w3a" "ta" "ph" "ps") "ch") "th") ("sylan" ("w3a" "ta" "ph" "ps") ("wa" "ph" "ps") "ch" "th" ("_3simpc" "ta" "ph" "ps") "_3adantl_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.) (theorem "_3adantl2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3adantl_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" ("w3a" "ph" "ta" "ps") "ch") "th") ("sylan" ("w3a" "ph" "ta" "ps") ("wa" "ph" "ps") "ch" "th" ("_3simpb" "ph" "ta" "ps") "_3adantl_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.) (theorem "_3adantl3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3adantl_1" ("wi" ("wa" ("wa" "ph" "ps") "ch") "th"))) (for) ("wi" ("wa" ("w3a" "ph" "ps" "ta") "ch") "th") ("sylan" ("w3a" "ph" "ps" "ta") ("wa" "ph" "ps") "ch" "th" ("_3simpa" "ph" "ps" "ta") "_3adantl_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.) (theorem "_3adantr1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3adantr_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ph" ("w3a" "ta" "ps" "ch")) "th") ("sylan2" ("w3a" "ta" "ps" "ch") "ph" ("wa" "ps" "ch") "th" ("_3simpc" "ta" "ps" "ch") "_3adantr_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.) (theorem "_3adantr2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3adantr_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ta" "ch")) "th") ("sylan2" ("w3a" "ps" "ta" "ch") "ph" ("wa" "ps" "ch") "th" ("_3simpb" "ps" "ta" "ch") "_3adantr_1")) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.) (theorem "_3adantr3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3adantr_1" ("wi" ("wa" "ph" ("wa" "ps" "ch")) "th"))) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ch" "ta")) "th") ("sylan2" ("w3a" "ps" "ch" "ta") "ph" ("wa" "ps" "ch") "th" ("_3simpa" "ps" "ch" "ta") "_3adantr_1")) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad4ant123" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "ta") "th") ("adantr" ("wa" ("wa" "ph" "ps") "ch") "th" "ta" ("_3expa" "ph" "ps" "ch" "th" "ad4ant3_1"))) ;; Obsolete version of ~ ad4ant123 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad4ant123OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "ta") "th") ("imp41" "ph" "ps" "ch" "ta" "th" ("a1ddd" "ph" "ps" "ch" "ta" "th" ("_3exp" "ph" "ps" "ch" "th" "ad4ant3_1")))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad4ant124" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ta") "ch") "th") ("adantlr" ("wa" "ph" "ps") "ch" "th" "ta" ("_3expa" "ph" "ps" "ch" "th" "ad4ant3_1"))) ;; Obsolete version of ~ ad4ant124 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad4ant124OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ta") "ch") "th") ("imp41" "ph" "ps" "ta" "ch" "th" ("a1dd" "ph" "ps" ("wi" "ch" "th") "ta" ("_3exp" "ph" "ps" "ch" "th" "ad4ant3_1")))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad4ant134" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ta") "ps") "ch") "th") ("adantllr" "ph" "ps" "ch" "th" "ta" ("_3expa" "ph" "ps" "ch" "th" "ad4ant3_1"))) ;; Obsolete version of ~ ad4ant134 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad4ant134OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ta") "ps") "ch") "th") ("imp41" "ph" "ta" "ps" "ch" "th" ("a1d" "ph" ("wi" "ps" ("wi" "ch" "th")) "ta" ("_3exp" "ph" "ps" "ch" "th" "ad4ant3_1")))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad4ant234" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" "ta" "ph") "ps") "ch") "th") ("adantlll" "ph" "ps" "ch" "th" "ta" ("_3expa" "ph" "ps" "ch" "th" "ad4ant3_1"))) ;; Obsolete version of ~ ad4ant234 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad4ant234OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" "ta" "ph") "ps") "ch") "th") ("imp41" "ta" "ph" "ps" "ch" "th" ("a1i" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))) "ta" ("_3exp" "ph" "ps" "ch" "th" "ad4ant3_1")))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "_3adant1l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" ("wa" "ta" "ph") "ps" "ch") "th") ("_3impa" ("wa" "ta" "ph") "ps" "ch" "th" ("ad4ant234" "ph" "ps" "ch" "th" "ta" "ad4ant3_1"))) ;; Obsolete version of ~ 3adant1l as of 23-Jun-2022. (Contributed by NM, 8-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3adant1lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" ("wa" "ta" "ph") "ps" "ch") "th") ("_3impb" ("wa" "ta" "ph") "ps" "ch" "th" ("adantll" "ph" ("wa" "ps" "ch") "th" "ta" ("_3expb" "ph" "ps" "ch" "th" "ad4ant3_1")))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "_3adant1r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" ("wa" "ph" "ta") "ps" "ch") "th") ("_3impa" ("wa" "ph" "ta") "ps" "ch" "th" ("ad4ant134" "ph" "ps" "ch" "th" "ta" "ad4ant3_1"))) ;; Obsolete version of ~ 3adant1r as of 23-Jun-2022. (Contributed by NM, 8-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3adant1rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" ("wa" "ph" "ta") "ps" "ch") "th") ("_3impb" ("wa" "ph" "ta") "ps" "ch" "th" ("adantlr" "ph" ("wa" "ps" "ch") "th" "ta" ("_3expb" "ph" "ps" "ch" "th" "ad4ant3_1")))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (theorem "_3adant2l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ph" ("wa" "ta" "ps") "ch") "th") ("_3com12" ("wa" "ta" "ps") "ph" "ch" "th" ("_3adant1l" "ps" "ph" "ch" "th" "ta" ("_3com12" "ph" "ps" "ch" "th" "ad4ant3_1")))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (theorem "_3adant2r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ph" ("wa" "ps" "ta") "ch") "th") ("_3com12" ("wa" "ps" "ta") "ph" "ch" "th" ("_3adant1r" "ps" "ph" "ch" "th" "ta" ("_3com12" "ph" "ps" "ch" "th" "ad4ant3_1")))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (theorem "_3adant3l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ph" "ps" ("wa" "ta" "ch")) "th") ("_3com13" ("wa" "ta" "ch") "ps" "ph" "th" ("_3adant1l" "ch" "ps" "ph" "th" "ta" ("_3com13" "ph" "ps" "ch" "th" "ad4ant3_1")))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (theorem "_3adant3r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ph" "ps" ("wa" "ch" "ta")) "th") ("_3com13" ("wa" "ch" "ta") "ps" "ph" "th" ("_3adant1r" "ch" "ps" "ph" "th" "ta" ("_3com13" "ph" "ps" "ch" "th" "ad4ant3_1")))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.) (theorem "_3adant3r1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" ("w3a" "ta" "ps" "ch")) "th") ("_3adantr1" "ph" "ps" "ch" "th" "ta" ("_3expb" "ph" "ps" "ch" "th" "ad4ant3_1"))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.) (theorem "_3adant3r2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ta" "ch")) "th") ("_3adantr2" "ph" "ps" "ch" "th" "ta" ("_3expb" "ph" "ps" "ch" "th" "ad4ant3_1"))) ;; Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.) (theorem "_3adant3r3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant3_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ch" "ta")) "th") ("_3adantr3" "ph" "ps" "ch" "th" "ta" ("_3expb" "ph" "ps" "ch" "th" "ad4ant3_1"))) ;; Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.) (theorem "_3ad2antl1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3ad2antl_1" ("wi" ("wa" "ph" "ch") "th"))) (for) ("wi" ("wa" ("w3a" "ph" "ps" "ta") "ch") "th") ("_3adantl2" "ph" "ta" "ch" "th" "ps" ("adantlr" "ph" "ch" "th" "ta" "_3ad2antl_1"))) ;; Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.) (theorem "_3ad2antl2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3ad2antl_1" ("wi" ("wa" "ph" "ch") "th"))) (for) ("wi" ("wa" ("w3a" "ps" "ph" "ta") "ch") "th") ("_3adantl1" "ph" "ta" "ch" "th" "ps" ("adantlr" "ph" "ch" "th" "ta" "_3ad2antl_1"))) ;; Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.) (theorem "_3ad2antl3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3ad2antl_1" ("wi" ("wa" "ph" "ch") "th"))) (for) ("wi" ("wa" ("w3a" "ps" "ta" "ph") "ch") "th") ("_3adantl1" "ta" "ph" "ch" "th" "ps" ("adantll" "ph" "ch" "th" "ta" "_3ad2antl_1"))) ;; Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.) (theorem "_3ad2antr1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3ad2antl_1" ("wi" ("wa" "ph" "ch") "th"))) (for) ("wi" ("wa" "ph" ("w3a" "ch" "ps" "ta")) "th") ("_3adantr3" "ph" "ch" "ps" "th" "ta" ("adantrr" "ph" "ch" "th" "ps" "_3ad2antl_1"))) ;; Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.) (theorem "_3ad2antr2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3ad2antl_1" ("wi" ("wa" "ph" "ch") "th"))) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ch" "ta")) "th") ("_3adantr3" "ph" "ps" "ch" "th" "ta" ("adantrl" "ph" "ch" "th" "ps" "_3ad2antl_1"))) ;; Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.) (theorem "_3ad2antr3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3ad2antl_1" ("wi" ("wa" "ph" "ch") "th"))) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ta" "ch")) "th") ("_3adantr1" "ph" "ta" "ch" "th" "ps" ("adantrl" "ph" "ch" "th" "ta" "_3ad2antl_1"))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad4ant13" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "th") "ps") "ta") "ch") ("adantr" ("wa" ("wa" "ph" "th") "ps") "ch" "ta" ("adantlr" "ph" "ps" "ch" "th" "ad4ant2_1"))) ;; Obsolete proof of ~ ad4ant13 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad4ant13OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "th") "ps") "ta") "ch") ("imp41" "ph" "th" "ps" "ta" "ch" ("a1d" "ph" ("wi" "ps" ("wi" "ta" "ch")) "th" ("a1dd" "ph" "ps" "ch" "ta" ("ex" "ph" "ps" "ch" "ad4ant2_1"))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad4ant14" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "th") "ta") "ps") "ch") ("adantlr" ("wa" "ph" "th") "ps" "ch" "ta" ("adantlr" "ph" "ps" "ch" "th" "ad4ant2_1"))) ;; Obsolete version of ~ ad4ant14 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad4ant14OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "th") "ta") "ps") "ch") ("imp41" "ph" "th" "ta" "ps" "ch" ("_2a1d" "ph" ("wi" "ps" "ch") "th" "ta" ("ex" "ph" "ps" "ch" "ad4ant2_1")))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad4ant23" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" "th" "ph") "ps") "ta") "ch") ("adantr" ("wa" ("wa" "th" "ph") "ps") "ch" "ta" ("adantll" "ph" "ps" "ch" "th" "ad4ant2_1"))) ;; Obsolete version of ~ ad4ant23 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad4ant23OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" "th" "ph") "ps") "ta") "ch") ("imp41" "th" "ph" "ps" "ta" "ch" ("a1i" ("wi" "ph" ("wi" "ps" ("wi" "ta" "ch"))) "th" ("a1dd" "ph" "ps" "ch" "ta" ("ex" "ph" "ps" "ch" "ad4ant2_1"))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad4ant24" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" "th" "ph") "ta") "ps") "ch") ("adantlr" ("wa" "th" "ph") "ps" "ch" "ta" ("adantll" "ph" "ps" "ch" "th" "ad4ant2_1"))) ;; Obsolete version of ~ ad4ant24 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad4ant24OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ad4ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" "th" "ph") "ta") "ps") "ch") ("imp41" "th" "ph" "ta" "ps" "ch" ("a1i13" "th" "ph" "ta" ("wi" "ps" "ch") ("ex" "ph" "ps" "ch" "ad4ant2_1")))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (theorem "ad5ant12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "th") "ta") "et") "ch") ("ad3antrrr" ("wa" "ph" "ps") "ch" "th" "ta" "et" "ad5ant2_1")) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad5ant13" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "th") "ps") "ta") "et") "ch") ("ad2antrr" ("wa" ("wa" "ph" "th") "ps") "ch" "ta" "et" ("adantlr" "ph" "ps" "ch" "th" "ad5ant2_1"))) ;; Obsolete version of ~ ad5ant13 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant13OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "th") "ps") "ta") "et") "ch") ("imp41" ("wa" "ph" "th") "ps" "ta" "et" "ch" ("imp" "ph" "th" ("wi" "ps" ("wi" "ta" ("wi" "et" "ch"))) ("com23" "ph" "ps" "th" ("wi" "ta" ("wi" "et" "ch")) ("com45" "ph" "ps" "th" "et" "ta" "ch" ("a1ddd" "ph" "ps" "th" "et" ("wi" "ta" "ch") ("_2a1dd" "ph" "ps" "ch" "th" "ta" ("ex" "ph" "ps" "ch" "ad5ant2_1")))))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad5ant14" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "th") "ta") "ps") "et") "ch") ("ad4ant13" ("wa" "ph" "th") "ps" "ch" "ta" "et" ("adantlr" "ph" "ps" "ch" "th" "ad5ant2_1"))) ;; Obsolete version of ~ ad5ant14 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant14OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "th") "ta") "ps") "et") "ch") ("imp41" ("wa" "ph" "th") "ta" "ps" "et" "ch" ("imp" "ph" "th" ("wi" "ta" ("wi" "ps" ("wi" "et" "ch"))) ("com34" "ph" "th" "ps" "ta" ("wi" "et" "ch") ("com23" "ph" "ps" "th" ("wi" "ta" ("wi" "et" "ch")) ("com45" "ph" "ps" "th" "et" "ta" "ch" ("a1ddd" "ph" "ps" "th" "et" ("wi" "ta" "ch") ("_2a1dd" "ph" "ps" "ch" "th" "ta" ("ex" "ph" "ps" "ch" "ad5ant2_1"))))))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad5ant15" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "th") "ta") "et") "ps") "ch") ("ad4ant14" ("wa" "ph" "th") "ps" "ch" "ta" "et" ("adantlr" "ph" "ps" "ch" "th" "ad5ant2_1"))) ;; Obsolete proof of ~ ad5ant15 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant15OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "th") "ta") "et") "ps") "ch") ("imp41" ("wa" "ph" "th") "ta" "et" "ps" "ch" ("imp" "ph" "th" ("wi" "ta" ("wi" "et" ("wi" "ps" "ch"))) ("com45" "ph" "th" "ta" "ps" "et" "ch" ("com34" "ph" "th" "ps" "ta" ("wi" "et" "ch") ("com23" "ph" "ps" "th" ("wi" "ta" ("wi" "et" "ch")) ("com45" "ph" "ps" "th" "et" "ta" "ch" ("a1ddd" "ph" "ps" "th" "et" ("wi" "ta" "ch") ("_2a1dd" "ph" "ps" "ch" "th" "ta" ("ex" "ph" "ps" "ch" "ad5ant2_1")))))))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad5ant23" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "th" "ph") "ps") "ta") "et") "ch") ("ad2antrr" ("wa" ("wa" "th" "ph") "ps") "ch" "ta" "et" ("adantll" "ph" "ps" "ch" "th" "ad5ant2_1"))) ;; Obsolete version of ~ ad5ant23 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant23OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "th" "ph") "ps") "ta") "et") "ch") ("imp41" ("wa" "th" "ph") "ps" "ta" "et" "ch" ("imp" "th" "ph" ("wi" "ps" ("wi" "ta" ("wi" "et" "ch"))) ("com3r" "ph" "ps" "th" ("wi" "ta" ("wi" "et" "ch")) ("com45" "ph" "ps" "th" "et" "ta" "ch" ("a1ddd" "ph" "ps" "th" "et" ("wi" "ta" "ch") ("_2a1dd" "ph" "ps" "ch" "th" "ta" ("ex" "ph" "ps" "ch" "ad5ant2_1")))))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad5ant24" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "th" "ph") "ta") "ps") "et") "ch") ("ad4ant13" ("wa" "th" "ph") "ps" "ch" "ta" "et" ("adantll" "ph" "ps" "ch" "th" "ad5ant2_1"))) ;; Obsolete version of ~ ad5ant24 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant24OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "th" "ph") "ta") "ps") "et") "ch") ("imp41" ("wa" "th" "ph") "ta" "ps" "et" "ch" ("imp" "th" "ph" ("wi" "ta" ("wi" "ps" ("wi" "et" "ch"))) ("com34" "th" "ph" "ps" "ta" ("wi" "et" "ch") ("com3r" "ph" "ps" "th" ("wi" "ta" ("wi" "et" "ch")) ("com45" "ph" "ps" "th" "et" "ta" "ch" ("a1ddd" "ph" "ps" "th" "et" ("wi" "ta" "ch") ("_2a1dd" "ph" "ps" "ch" "th" "ta" ("ex" "ph" "ps" "ch" "ad5ant2_1"))))))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad5ant25" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "th" "ph") "ta") "et") "ps") "ch") ("ad4ant14" ("wa" "th" "ph") "ps" "ch" "ta" "et" ("adantll" "ph" "ps" "ch" "th" "ad5ant2_1"))) ;; Obsolete version of ~ ad5ant25 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant25OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2_1" ("wi" ("wa" "ph" "ps") "ch"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "th" "ph") "ta") "et") "ps") "ch") ("imp41" ("wa" "th" "ph") "ta" "et" "ps" "ch" ("imp" "th" "ph" ("wi" "ta" ("wi" "et" ("wi" "ps" "ch"))) ("com45" "th" "ph" "ta" "ps" "et" "ch" ("com34" "th" "ph" "ps" "ta" ("wi" "et" "ch") ("com3r" "ph" "ps" "th" ("wi" "ta" ("wi" "et" "ch")) ("com45" "ph" "ps" "th" "et" "ta" "ch" ("a1ddd" "ph" "ps" "th" "et" ("wi" "ta" "ch") ("_2a1dd" "ph" "ps" "ch" "th" "ta" ("ex" "ph" "ps" "ch" "ad5ant2_1")))))))))) ;; Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpl1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ph") ("_3ad2antl1" "ph" "ps" "th" "ph" "ch" ("simpl" "ph" "th"))) ;; Obsolete version of ~ simpl1 as of 23-Jun-2022. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpl1OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ph") ("adantr" ("w3a" "ph" "ps" "ch") "ph" "th" ("simp1" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpl2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ps") ("_3ad2antl2" "ps" "ph" "th" "ps" "ch" ("simpl" "ps" "th"))) ;; Obsolete version of ~ simpl2 as of 23-Jun-2022. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpl2OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ps") ("adantr" ("w3a" "ph" "ps" "ch") "ps" "th" ("simp2" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpl3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ch") ("_3ad2antl3" "ch" "ph" "th" "ch" "ps" ("simpl" "ch" "th"))) ;; Obsolete version of ~ simpl3 as of 23-Jun-2022. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpl3OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ch") ("adantr" ("w3a" "ph" "ps" "ch") "ch" "th" ("simp3" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpr1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ps") ("_3ad2antr1" "ph" "ch" "ps" "ps" "th" ("simpr" "ph" "ps"))) ;; Obsolete version of ~ simpr1 as of 23-Jun-2022. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpr1OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ps") ("adantl" ("w3a" "ps" "ch" "th") "ps" "ph" ("simp1" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpr2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ch") ("_3ad2antr2" "ph" "ps" "ch" "ch" "th" ("simpr" "ph" "ch"))) ;; Obsolete version of ~ simpr2 as of 23-Jun-2022. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpr2OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ch") ("adantl" ("w3a" "ps" "ch" "th") "ch" "ph" ("simp2" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpr3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "th") ("_3ad2antr3" "ph" "ps" "th" "th" "ch" ("simpr" "ph" "th"))) ;; Obsolete version of ~ simpr3 as of 23-Jun-2022. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpr3OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "th") ("adantl" ("w3a" "ps" "ch" "th") "th" "ph" ("simp3" "ps" "ch" "th"))) ;; Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) (theorem "simp1l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" "ph" "ps") "ch" "th") "ph") ("_3ad2ant1" ("wa" "ph" "ps") "ch" "ph" "th" ("simpl" "ph" "ps"))) ;; Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) (theorem "simp1r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" "ph" "ps") "ch" "th") "ps") ("_3ad2ant1" ("wa" "ph" "ps") "ch" "ps" "th" ("simpr" "ph" "ps"))) ;; Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) (theorem "simp2l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" ("wa" "ps" "ch") "th") "ps") ("_3ad2ant2" ("wa" "ps" "ch") "ph" "ps" "th" ("simpl" "ps" "ch"))) ;; Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) (theorem "simp2r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" ("wa" "ps" "ch") "th") "ch") ("_3ad2ant2" ("wa" "ps" "ch") "ph" "ch" "th" ("simpr" "ps" "ch"))) ;; Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) (theorem "simp3l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" ("wa" "ch" "th")) "ch") ("_3ad2ant3" ("wa" "ch" "th") "ph" "ch" "ps" ("simpl" "ch" "th"))) ;; Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) (theorem "simp3r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" ("wa" "ch" "th")) "th") ("_3ad2ant3" ("wa" "ch" "th") "ph" "th" "ps" ("simpr" "ch" "th"))) ;; Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) (theorem "simp11" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ph") ("_3ad2ant1" ("w3a" "ph" "ps" "ch") "th" "ph" "ta" ("simp1" "ph" "ps" "ch"))) ;; Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) (theorem "simp12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ps") ("_3ad2ant1" ("w3a" "ph" "ps" "ch") "th" "ps" "ta" ("simp2" "ph" "ps" "ch"))) ;; Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) (theorem "simp13" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ch") ("_3ad2ant1" ("w3a" "ph" "ps" "ch") "th" "ch" "ta" ("simp3" "ph" "ps" "ch"))) ;; Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) (theorem "simp21" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" ("w3a" "ps" "ch" "th") "ta") "ps") ("_3ad2ant2" ("w3a" "ps" "ch" "th") "ph" "ps" "ta" ("simp1" "ps" "ch" "th"))) ;; Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) (theorem "simp22" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" ("w3a" "ps" "ch" "th") "ta") "ch") ("_3ad2ant2" ("w3a" "ps" "ch" "th") "ph" "ch" "ta" ("simp2" "ps" "ch" "th"))) ;; Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) (theorem "simp23" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" ("w3a" "ps" "ch" "th") "ta") "th") ("_3ad2ant2" ("w3a" "ps" "ch" "th") "ph" "th" "ta" ("simp3" "ps" "ch" "th"))) ;; Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) (theorem "simp31" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" ("w3a" "ch" "th" "ta")) "ch") ("_3ad2ant3" ("w3a" "ch" "th" "ta") "ph" "ch" "ps" ("simp1" "ch" "th" "ta"))) ;; Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) (theorem "simp32" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" ("w3a" "ch" "th" "ta")) "th") ("_3ad2ant3" ("w3a" "ch" "th" "ta") "ph" "th" "ps" ("simp2" "ch" "th" "ta"))) ;; Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.) (theorem "simp33" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "ph" "ps" ("w3a" "ch" "th" "ta")) "ta") ("_3ad2ant3" ("w3a" "ch" "th" "ta") "ph" "ta" "ps" ("simp3" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpll1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta") "ph") ("ad2antrr" ("w3a" "ph" "ps" "ch") "ph" "th" "ta" ("simp1" "ph" "ps" "ch"))) ;; Obsolete version of ~ simpll1 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpll1OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta") "ph") ("adantr" ("wa" ("w3a" "ph" "ps" "ch") "th") "ph" "ta" ("simpl1" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpll2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta") "ps") ("ad2antrr" ("w3a" "ph" "ps" "ch") "ps" "th" "ta" ("simp2" "ph" "ps" "ch"))) ;; Obsolete version of ~ simpll2 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpll2OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta") "ps") ("adantr" ("wa" ("w3a" "ph" "ps" "ch") "th") "ps" "ta" ("simpl2" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpll3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta") "ch") ("ad2antrr" ("w3a" "ph" "ps" "ch") "ch" "th" "ta" ("simp3" "ph" "ps" "ch"))) ;; Obsolete version of ~ simpll3 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpll3OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta") "ch") ("adantr" ("wa" ("w3a" "ph" "ps" "ch") "th") "ch" "ta" ("simpl3" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simplr1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta") "ph") ("ad2antlr" ("w3a" "ph" "ps" "ch") "ph" "th" "ta" ("simp1" "ph" "ps" "ch"))) ;; Obsolete version of ~ simplr1 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simplr1OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta") "ph") ("adantr" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ph" "ta" ("simpr1" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simplr2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta") "ps") ("ad2antlr" ("w3a" "ph" "ps" "ch") "ps" "th" "ta" ("simp2" "ph" "ps" "ch"))) ;; Obsolete version of ~ simplr2 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simplr2OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta") "ps") ("adantr" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ps" "ta" ("simpr2" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simplr3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta") "ch") ("ad2antlr" ("w3a" "ph" "ps" "ch") "ch" "th" "ta" ("simp3" "ph" "ps" "ch"))) ;; Obsolete version of ~ simplr3 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simplr3OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta") "ch") ("adantr" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ch" "ta" ("simpr3" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simprl1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" ("w3a" "ph" "ps" "ch") "th")) "ph") ("ad2antrl" ("w3a" "ph" "ps" "ch") "ph" "ta" "th" ("simp1" "ph" "ps" "ch"))) ;; Obsolete version of ~ simprl1 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simprl1OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" ("w3a" "ph" "ps" "ch") "th")) "ph") ("adantl" ("wa" ("w3a" "ph" "ps" "ch") "th") "ph" "ta" ("simpl1" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simprl2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" ("w3a" "ph" "ps" "ch") "th")) "ps") ("ad2antrl" ("w3a" "ph" "ps" "ch") "ps" "ta" "th" ("simp2" "ph" "ps" "ch"))) ;; Obsolete version of ~ simprl2 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simprl2OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" ("w3a" "ph" "ps" "ch") "th")) "ps") ("adantl" ("wa" ("w3a" "ph" "ps" "ch") "th") "ps" "ta" ("simpl2" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simprl3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" ("w3a" "ph" "ps" "ch") "th")) "ch") ("ad2antrl" ("w3a" "ph" "ps" "ch") "ch" "ta" "th" ("simp3" "ph" "ps" "ch"))) ;; Obsolete version of ~ simprl3 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simprl3OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" ("w3a" "ph" "ps" "ch") "th")) "ch") ("adantl" ("wa" ("w3a" "ph" "ps" "ch") "th") "ch" "ta" ("simpl3" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simprr1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" "th" ("w3a" "ph" "ps" "ch"))) "ph") ("ad2antll" ("w3a" "ph" "ps" "ch") "ph" "ta" "th" ("simp1" "ph" "ps" "ch"))) ;; Obsolete version of ~ simprr1 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simprr1OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" "th" ("w3a" "ph" "ps" "ch"))) "ph") ("adantl" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ph" "ta" ("simpr1" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simprr2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" "th" ("w3a" "ph" "ps" "ch"))) "ps") ("ad2antll" ("w3a" "ph" "ps" "ch") "ps" "ta" "th" ("simp2" "ph" "ps" "ch"))) ;; Obsolete version of ~ simprr2 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simprr2OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" "th" ("w3a" "ph" "ps" "ch"))) "ps") ("adantl" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ps" "ta" ("simpr2" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simprr3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" "th" ("w3a" "ph" "ps" "ch"))) "ch") ("ad2antll" ("w3a" "ph" "ps" "ch") "ch" "ta" "th" ("simp3" "ph" "ps" "ch"))) ;; Obsolete version of ~ simprr3 as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simprr3OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("wa" "th" ("w3a" "ph" "ps" "ch"))) "ch") ("adantl" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ch" "ta" ("simpr3" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpl1l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta") "ph") ("_3ad2antl1" ("wa" "ph" "ps") "ch" "ta" "ph" "th" ("simpll" "ph" "ps" "ta"))) ;; Obsolete version of ~ simpl1l as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpl1lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta") "ph") ("adantr" ("w3a" ("wa" "ph" "ps") "ch" "th") "ph" "ta" ("simp1l" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpl1r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta") "ps") ("_3ad2antl1" ("wa" "ph" "ps") "ch" "ta" "ps" "th" ("simplr" "ph" "ps" "ta"))) ;; Obsolete version of ~ simpl1r as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpl1rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta") "ps") ("adantr" ("w3a" ("wa" "ph" "ps") "ch" "th") "ps" "ta" ("simp1r" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpl2l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta") "ph") ("_3ad2antl2" ("wa" "ph" "ps") "ch" "ta" "ph" "th" ("simpll" "ph" "ps" "ta"))) ;; Obsolete version of ~ simpl2l as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpl2lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta") "ph") ("adantr" ("w3a" "ch" ("wa" "ph" "ps") "th") "ph" "ta" ("simp2l" "ch" "ph" "ps" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpl2r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta") "ps") ("_3ad2antl2" ("wa" "ph" "ps") "ch" "ta" "ps" "th" ("simplr" "ph" "ps" "ta"))) ;; Obsolete version of ~ simpl2r as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpl2rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta") "ps") ("adantr" ("w3a" "ch" ("wa" "ph" "ps") "th") "ps" "ta" ("simp2r" "ch" "ph" "ps" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpl3l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta") "ph") ("_3ad2antl3" ("wa" "ph" "ps") "ch" "ta" "ph" "th" ("simpll" "ph" "ps" "ta"))) ;; Obsolete version of ~ simpl3l as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpl3lOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta") "ph") ("adantr" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ph" "ta" ("simp3l" "ch" "th" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "simpl3r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta") "ps") ("_3ad2antl3" ("wa" "ph" "ps") "ch" "ta" "ps" "th" ("simplr" "ph" "ps" "ta"))) ;; Obsolete version of ~ simpl3r as of 23-Jun-2022. (Contributed by NM, 9-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "simpl3rOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta") "ps") ("adantr" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ps" "ta" ("simp3r" "ch" "th" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr1l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("w3a" ("wa" "ph" "ps") "ch" "th")) "ph") ("adantl" ("w3a" ("wa" "ph" "ps") "ch" "th") "ph" "ta" ("simp1l" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr1r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("w3a" ("wa" "ph" "ps") "ch" "th")) "ps") ("adantl" ("w3a" ("wa" "ph" "ps") "ch" "th") "ps" "ta" ("simp1r" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr2l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("w3a" "ch" ("wa" "ph" "ps") "th")) "ph") ("adantl" ("w3a" "ch" ("wa" "ph" "ps") "th") "ph" "ta" ("simp2l" "ch" "ph" "ps" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr2r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("w3a" "ch" ("wa" "ph" "ps") "th")) "ps") ("adantl" ("w3a" "ch" ("wa" "ph" "ps") "th") "ps" "ta" ("simp2r" "ch" "ph" "ps" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr3l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("w3a" "ch" "th" ("wa" "ph" "ps"))) "ph") ("adantl" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ph" "ta" ("simp3l" "ch" "th" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr3r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("wa" "ta" ("w3a" "ch" "th" ("wa" "ph" "ps"))) "ps") ("adantl" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ps" "ta" ("simp3r" "ch" "th" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp1ll" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" ("wa" "ph" "ps") "ch") "th" "ta") "ph") ("_3ad2ant1" ("wa" ("wa" "ph" "ps") "ch") "th" "ph" "ta" ("simpll" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp1lr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" ("wa" "ph" "ps") "ch") "th" "ta") "ps") ("_3ad2ant1" ("wa" ("wa" "ph" "ps") "ch") "th" "ps" "ta" ("simplr" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp1rl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" "ch" ("wa" "ph" "ps")) "th" "ta") "ph") ("_3ad2ant1" ("wa" "ch" ("wa" "ph" "ps")) "th" "ph" "ta" ("simprl" "ch" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp1rr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" "ch" ("wa" "ph" "ps")) "th" "ta") "ps") ("_3ad2ant1" ("wa" "ch" ("wa" "ph" "ps")) "th" "ps" "ta" ("simprr" "ch" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp2ll" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "th" ("wa" ("wa" "ph" "ps") "ch") "ta") "ph") ("_3ad2ant2" ("wa" ("wa" "ph" "ps") "ch") "th" "ph" "ta" ("simpll" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp2lr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "th" ("wa" ("wa" "ph" "ps") "ch") "ta") "ps") ("_3ad2ant2" ("wa" ("wa" "ph" "ps") "ch") "th" "ps" "ta" ("simplr" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp2rl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "th" ("wa" "ch" ("wa" "ph" "ps")) "ta") "ph") ("_3ad2ant2" ("wa" "ch" ("wa" "ph" "ps")) "th" "ph" "ta" ("simprl" "ch" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp2rr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "th" ("wa" "ch" ("wa" "ph" "ps")) "ta") "ps") ("_3ad2ant2" ("wa" "ch" ("wa" "ph" "ps")) "th" "ps" "ta" ("simprr" "ch" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp3ll" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "th" "ta" ("wa" ("wa" "ph" "ps") "ch")) "ph") ("_3ad2ant3" ("wa" ("wa" "ph" "ps") "ch") "th" "ph" "ta" ("simpll" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp3lr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "th" "ta" ("wa" ("wa" "ph" "ps") "ch")) "ps") ("_3ad2ant3" ("wa" ("wa" "ph" "ps") "ch") "th" "ps" "ta" ("simplr" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp3rl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "th" "ta" ("wa" "ch" ("wa" "ph" "ps"))) "ph") ("_3ad2ant3" ("wa" "ch" ("wa" "ph" "ps")) "th" "ph" "ta" ("simprl" "ch" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp3rr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" ("w3a" "th" "ta" ("wa" "ch" ("wa" "ph" "ps"))) "ps") ("_3ad2ant3" ("wa" "ch" ("wa" "ph" "ps")) "th" "ps" "ta" ("simprr" "ch" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpl11" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et") "ph") ("adantr" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ph" "et" ("simp11" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpl12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et") "ps") ("adantr" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ps" "et" ("simp12" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpl13" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et") "ch") ("adantr" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ch" "et" ("simp13" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpl21" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et") "ph") ("adantr" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "ph" "et" ("simp21" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpl22" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et") "ps") ("adantr" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "ps" "et" ("simp22" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpl23" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et") "ch") ("adantr" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "ch" "et" ("simp23" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpl31" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et") "ph") ("adantr" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "ph" "et" ("simp31" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpl32" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et") "ps") ("adantr" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "ps" "et" ("simp32" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpl33" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et") "ch") ("adantr" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "ch" "et" ("simp33" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr11" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" "et" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta")) "ph") ("adantl" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ph" "et" ("simp11" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" "et" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta")) "ps") ("adantl" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ps" "et" ("simp12" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr13" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" "et" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta")) "ch") ("adantl" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ch" "et" ("simp13" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr21" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" "et" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta")) "ph") ("adantl" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "ph" "et" ("simp21" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr22" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" "et" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta")) "ps") ("adantl" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "ps" "et" ("simp22" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr23" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" "et" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta")) "ch") ("adantl" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "ch" "et" ("simp23" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr31" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" "et" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch"))) "ph") ("adantl" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "ph" "et" ("simp31" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr32" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" "et" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch"))) "ps") ("adantl" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "ps" "et" ("simp32" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simpr33" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("wa" "et" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch"))) "ch") ("adantl" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "ch" "et" ("simp33" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp1l1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "et") "ph") ("_3ad2ant1" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "ph" "et" ("simpl1" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp1l2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "et") "ps") ("_3ad2ant1" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "ps" "et" ("simpl2" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp1l3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "et") "ch") ("_3ad2ant1" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "ch" "et" ("simpl3" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp1r1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "et") "ph") ("_3ad2ant1" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "ph" "et" ("simpr1" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp1r2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "et") "ps") ("_3ad2ant1" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "ps" "et" ("simpr2" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp1r3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "et") "ch") ("_3ad2ant1" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "ch" "et" ("simpr3" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp2l1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("wa" ("w3a" "ph" "ps" "ch") "th") "et") "ph") ("_3ad2ant2" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "ph" "et" ("simpl1" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp2l2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("wa" ("w3a" "ph" "ps" "ch") "th") "et") "ps") ("_3ad2ant2" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "ps" "et" ("simpl2" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp2l3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("wa" ("w3a" "ph" "ps" "ch") "th") "et") "ch") ("_3ad2ant2" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "ch" "et" ("simpl3" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp2r1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("wa" "th" ("w3a" "ph" "ps" "ch")) "et") "ph") ("_3ad2ant2" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "ph" "et" ("simpr1" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp2r2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("wa" "th" ("w3a" "ph" "ps" "ch")) "et") "ps") ("_3ad2ant2" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "ps" "et" ("simpr2" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp2r3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("wa" "th" ("w3a" "ph" "ps" "ch")) "et") "ch") ("_3ad2ant2" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "ch" "et" ("simpr3" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp3l1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("wa" ("w3a" "ph" "ps" "ch") "th")) "ph") ("_3ad2ant3" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "ph" "et" ("simpl1" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp3l2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("wa" ("w3a" "ph" "ps" "ch") "th")) "ps") ("_3ad2ant3" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "ps" "et" ("simpl2" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp3l3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("wa" ("w3a" "ph" "ps" "ch") "th")) "ch") ("_3ad2ant3" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta" "ch" "et" ("simpl3" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp3r1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("wa" "th" ("w3a" "ph" "ps" "ch"))) "ph") ("_3ad2ant3" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "ph" "et" ("simpr1" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp3r2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("wa" "th" ("w3a" "ph" "ps" "ch"))) "ps") ("_3ad2ant3" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "ps" "et" ("simpr2" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp3r3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("wa" "th" ("w3a" "ph" "ps" "ch"))) "ch") ("_3ad2ant3" ("wa" "th" ("w3a" "ph" "ps" "ch")) "ta" "ch" "et" ("simpr3" "th" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp11l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta" "et") "ph") ("_3ad2ant1" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta" "ph" "et" ("simp1l" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp11r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta" "et") "ps") ("_3ad2ant1" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta" "ps" "et" ("simp1r" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp12l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta" "et") "ph") ("_3ad2ant1" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta" "ph" "et" ("simp2l" "ch" "ph" "ps" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp12r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta" "et") "ps") ("_3ad2ant1" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta" "ps" "et" ("simp2r" "ch" "ph" "ps" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp13l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta" "et") "ph") ("_3ad2ant1" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta" "ph" "et" ("simp3l" "ch" "th" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp13r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta" "et") "ps") ("_3ad2ant1" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta" "ps" "et" ("simp3r" "ch" "th" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp21l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("w3a" ("wa" "ph" "ps") "ch" "th") "et") "ph") ("_3ad2ant2" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta" "ph" "et" ("simp1l" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp21r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("w3a" ("wa" "ph" "ps") "ch" "th") "et") "ps") ("_3ad2ant2" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta" "ps" "et" ("simp1r" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp22l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("w3a" "ch" ("wa" "ph" "ps") "th") "et") "ph") ("_3ad2ant2" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta" "ph" "et" ("simp2l" "ch" "ph" "ps" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp22r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("w3a" "ch" ("wa" "ph" "ps") "th") "et") "ps") ("_3ad2ant2" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta" "ps" "et" ("simp2r" "ch" "ph" "ps" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp23l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("w3a" "ch" "th" ("wa" "ph" "ps")) "et") "ph") ("_3ad2ant2" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta" "ph" "et" ("simp3l" "ch" "th" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp23r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" ("w3a" "ch" "th" ("wa" "ph" "ps")) "et") "ps") ("_3ad2ant2" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta" "ps" "et" ("simp3r" "ch" "th" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp31l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("w3a" ("wa" "ph" "ps") "ch" "th")) "ph") ("_3ad2ant3" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta" "ph" "et" ("simp1l" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp31r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("w3a" ("wa" "ph" "ps") "ch" "th")) "ps") ("_3ad2ant3" ("w3a" ("wa" "ph" "ps") "ch" "th") "ta" "ps" "et" ("simp1r" "ph" "ps" "ch" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp32l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("w3a" "ch" ("wa" "ph" "ps") "th")) "ph") ("_3ad2ant3" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta" "ph" "et" ("simp2l" "ch" "ph" "ps" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp32r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("w3a" "ch" ("wa" "ph" "ps") "th")) "ps") ("_3ad2ant3" ("w3a" "ch" ("wa" "ph" "ps") "th") "ta" "ps" "et" ("simp2r" "ch" "ph" "ps" "th"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp33l" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("w3a" "ch" "th" ("wa" "ph" "ps"))) "ph") ("_3ad2ant3" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta" "ph" "et" ("simp3l" "ch" "th" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp33r" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wi" ("w3a" "ta" "et" ("w3a" "ch" "th" ("wa" "ph" "ps"))) "ps") ("_3ad2ant3" ("w3a" "ch" "th" ("wa" "ph" "ps")) "ta" "ps" "et" ("simp3r" "ch" "th" "ph" "ps"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp111" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ze") "ph") ("_3ad2ant1" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ph" "ze" ("simp11" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp112" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ze") "ps") ("_3ad2ant1" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ps" "ze" ("simp12" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp113" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ze") "ch") ("_3ad2ant1" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ch" "ze" ("simp13" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp121" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ze") "ph") ("_3ad2ant1" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ph" "ze" ("simp21" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp122" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ze") "ps") ("_3ad2ant1" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ps" "ze" ("simp22" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp123" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ze") "ch") ("_3ad2ant1" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ch" "ze" ("simp23" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp131" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ze") "ph") ("_3ad2ant1" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ph" "ze" ("simp31" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp132" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ze") "ps") ("_3ad2ant1" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ps" "ze" ("simp32" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp133" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ze") "ch") ("_3ad2ant1" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ch" "ze" ("simp33" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp211" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ze") "ph") ("_3ad2ant2" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ph" "ze" ("simp11" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp212" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ze") "ps") ("_3ad2ant2" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ps" "ze" ("simp12" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp213" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "ze") "ch") ("_3ad2ant2" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ch" "ze" ("simp13" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp221" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "ze") "ph") ("_3ad2ant2" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ph" "ze" ("simp21" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp222" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "ze") "ps") ("_3ad2ant2" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ps" "ze" ("simp22" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp223" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "ze") "ch") ("_3ad2ant2" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ch" "ze" ("simp23" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp231" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "ze") "ph") ("_3ad2ant2" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ph" "ze" ("simp31" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp232" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "ze") "ps") ("_3ad2ant2" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ps" "ze" ("simp32" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp233" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "ze") "ch") ("_3ad2ant2" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ch" "ze" ("simp33" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp311" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" "ze" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta")) "ph") ("_3ad2ant3" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ph" "ze" ("simp11" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp312" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" "ze" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta")) "ps") ("_3ad2ant3" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ps" "ze" ("simp12" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp313" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" "ze" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta")) "ch") ("_3ad2ant3" ("w3a" ("w3a" "ph" "ps" "ch") "th" "ta") "et" "ch" "ze" ("simp13" "ph" "ps" "ch" "th" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp321" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" "ze" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta")) "ph") ("_3ad2ant3" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ph" "ze" ("simp21" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp322" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" "ze" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta")) "ps") ("_3ad2ant3" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ps" "ze" ("simp22" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp323" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" "ze" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta")) "ch") ("_3ad2ant3" ("w3a" "th" ("w3a" "ph" "ps" "ch") "ta") "et" "ch" "ze" ("simp23" "th" "ph" "ps" "ch" "ta"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp331" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" "ze" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch"))) "ph") ("_3ad2ant3" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ph" "ze" ("simp31" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp332" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" "ze" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch"))) "ps") ("_3ad2ant3" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ps" "ze" ("simp32" "th" "ta" "ph" "ps" "ch"))) ;; Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (theorem "simp333" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for) (for) ("wi" ("w3a" "et" "ze" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch"))) "ch") ("_3ad2ant3" ("w3a" "th" "ta" ("w3a" "ph" "ps" "ch")) "et" "ch" "ze" ("simp33" "th" "ta" "ph" "ps" "ch"))) ;; Remove a hypothesis from the second member of a biimplication. (Contributed by FL, 22-Jul-2008.) (theorem "_3anibar" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3anibar_1" ("wi" ("w3a" "ph" "ps" "ch") ("wb" "th" ("wa" "ch" "ta"))))) (for) ("wi" ("w3a" "ph" "ps" "ch") ("wb" "th" "ta")) ("mpbirand" ("w3a" "ph" "ps" "ch") "th" "ch" "ta" ("simp3" "ph" "ps" "ch") "_3anibar_1")) ;; Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.) (theorem "_3mix1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" "ph" ("w3o" "ph" "ps" "ch")) ("sylibr" "ph" ("wo" "ph" ("wo" "ps" "ch")) ("w3o" "ph" "ps" "ch") ("orc" "ph" ("wo" "ps" "ch")) ("_3orass" "ph" "ps" "ch"))) ;; Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.) (theorem "_3mix2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" "ph" ("w3o" "ps" "ph" "ch")) ("sylibr" "ph" ("w3o" "ph" "ch" "ps") ("w3o" "ps" "ph" "ch") ("_3mix1" "ph" "ch" "ps") ("_3orrot" "ps" "ph" "ch"))) ;; Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.) (theorem "_3mix3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" "ph" ("w3o" "ps" "ch" "ph")) ("sylib" "ph" ("w3o" "ph" "ps" "ch") ("w3o" "ps" "ch" "ph") ("_3mix1" "ph" "ps" "ch") ("_3orrot" "ph" "ps" "ch"))) ;; Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.) (theorem "_3mix1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_3mixi_1" "ph")) (for) ("w3o" "ph" "ps" "ch") ("ax_mp" "ph" ("w3o" "ph" "ps" "ch") "_3mixi_1" ("_3mix1" "ph" "ps" "ch"))) ;; Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.) (theorem "_3mix2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_3mixi_1" "ph")) (for) ("w3o" "ps" "ph" "ch") ("ax_mp" "ph" ("w3o" "ps" "ph" "ch") "_3mixi_1" ("_3mix2" "ph" "ps" "ch"))) ;; Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.) (theorem "_3mix3i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_3mixi_1" "ph")) (for) ("w3o" "ps" "ch" "ph") ("ax_mp" "ph" ("w3o" "ps" "ch" "ph") "_3mixi_1" ("_3mix3" "ph" "ps" "ch"))) ;; Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.) (theorem "_3mix1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3mixd_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("w3o" "ps" "ch" "th")) ("syl" "ph" "ps" ("w3o" "ps" "ch" "th") "_3mixd_1" ("_3mix1" "ps" "ch" "th"))) ;; Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.) (theorem "_3mix2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3mixd_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("w3o" "ch" "ps" "th")) ("syl" "ph" "ps" ("w3o" "ch" "ps" "th") "_3mixd_1" ("_3mix2" "ps" "ch" "th"))) ;; Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.) (theorem "_3mix3d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3mixd_1" ("wi" "ph" "ps"))) (for) ("wi" "ph" ("w3o" "ch" "th" "ps")) ("syl" "ph" "ps" ("w3o" "ch" "th" "ps") "_3mixd_1" ("_3mix3" "ps" "ch" "th"))) ;; Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.) (theorem "_3pm3_2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_3pm3_2i_1" "ph") ("_3pm3_2i_2" "ps") ("_3pm3_2i_3" "ch")) (for) ("w3a" "ph" "ps" "ch") ("mpbir2an" ("w3a" "ph" "ps" "ch") ("wa" "ph" "ps") "ch" ("pm3_2i" "ph" "ps" "_3pm3_2i_1" "_3pm3_2i_2") "_3pm3_2i_3" ("df_3an" "ph" "ps" "ch"))) ;; Version of ~ pm3.2 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Kyle Wyonch, 24-Apr-2021.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) (theorem "pm3_2an3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("w3a" "ph" "ps" "ch")))) ("_3exp" "ph" "ps" "ch" ("w3a" "ph" "ps" "ch") ("id" ("w3a" "ph" "ps" "ch")))) ;; Obsolete version of ~ pm3.2an3 as of 21-Jun-2022. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Kyle Wyonch, 24-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "pm3_2an3OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("w3a" "ph" "ps" "ch")))) ("exp31" "ph" "ps" "ch" ("w3a" "ph" "ps" "ch") ("biimpri" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") "ch") ("df_3an" "ph" "ps" "ch")))) ;; Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.) (theorem "_3jca" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3jca_1" ("wi" "ph" "ps")) ("_3jca_2" ("wi" "ph" "ch")) ("_3jca_3" ("wi" "ph" "th"))) (for) ("wi" "ph" ("w3a" "ps" "ch" "th")) ("sylibr" "ph" ("wa" ("wa" "ps" "ch") "th") ("w3a" "ps" "ch" "th") ("jca31" "ph" "ps" "ch" "th" "_3jca_1" "_3jca_2" "_3jca_3") ("df_3an" "ps" "ch" "th"))) ;; Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.) (theorem "_3jcad" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3jcad_1" ("wi" "ph" ("wi" "ps" "ch"))) ("_3jcad_2" ("wi" "ph" ("wi" "ps" "th"))) ("_3jcad_3" ("wi" "ph" ("wi" "ps" "ta")))) (for) ("wi" "ph" ("wi" "ps" ("w3a" "ch" "th" "ta"))) ("ex" "ph" "ps" ("w3a" "ch" "th" "ta") ("_3jca" ("wa" "ph" "ps") "ch" "th" "ta" ("imp" "ph" "ps" "ch" "_3jcad_1") ("imp" "ph" "ps" "th" "_3jcad_2") ("imp" "ph" "ps" "ta" "_3jcad_3")))) ;; Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.) (theorem "mpbir3an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpbir3an_1" "ps") ("mpbir3an_2" "ch") ("mpbir3an_3" "th") ("mpbir3an_4" ("wb" "ph" ("w3a" "ps" "ch" "th")))) (for) "ph" ("mpbir" "ph" ("w3a" "ps" "ch" "th") ("_3pm3_2i" "ps" "ch" "th" "mpbir3an_1" "mpbir3an_2" "mpbir3an_3") "mpbir3an_4")) ;; Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.) (theorem "mpbir3and" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mpbir3and_1" ("wi" "ph" "ch")) ("mpbir3and_2" ("wi" "ph" "th")) ("mpbir3and_3" ("wi" "ph" "ta")) ("mpbir3and_4" ("wi" "ph" ("wb" "ps" ("w3a" "ch" "th" "ta"))))) (for) ("wi" "ph" "ps") ("mpbird" "ph" "ps" ("w3a" "ch" "th" "ta") ("_3jca" "ph" "ch" "th" "ta" "mpbir3and_1" "mpbir3and_2" "mpbir3and_3") "mpbir3and_4")) ;; Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.) (theorem "syl3anbrc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl3anbrc_1" ("wi" "ph" "ps")) ("syl3anbrc_2" ("wi" "ph" "ch")) ("syl3anbrc_3" ("wi" "ph" "th")) ("syl3anbrc_4" ("wb" "ta" ("w3a" "ps" "ch" "th")))) (for) ("wi" "ph" "ta") ("sylibr" "ph" ("w3a" "ps" "ch" "th") "ta" ("_3jca" "ph" "ps" "ch" "th" "syl3anbrc_1" "syl3anbrc_2" "syl3anbrc_3") "syl3anbrc_4")) ;; Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.) (theorem "_3anim123i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("_3anim123i_1" ("wi" "ph" "ps")) ("_3anim123i_2" ("wi" "ch" "th")) ("_3anim123i_3" ("wi" "ta" "et"))) (for) ("wi" ("w3a" "ph" "ch" "ta") ("w3a" "ps" "th" "et")) ("_3jca" ("w3a" "ph" "ch" "ta") "ps" "th" "et" ("_3ad2ant1" "ph" "ch" "ps" "ta" "_3anim123i_1") ("_3ad2ant2" "ch" "ph" "th" "ta" "_3anim123i_2") ("_3ad2ant3" "ta" "ph" "et" "ch" "_3anim123i_3"))) ;; Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.) (theorem "_3anim1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3animi_1" ("wi" "ph" "ps"))) (for) ("wi" ("w3a" "ph" "ch" "th") ("w3a" "ps" "ch" "th")) ("_3anim123i" "ph" "ps" "ch" "ch" "th" "th" "_3animi_1" ("id" "ch") ("id" "th"))) ;; Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.) (theorem "_3anim2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3animi_1" ("wi" "ph" "ps"))) (for) ("wi" ("w3a" "ch" "ph" "th") ("w3a" "ch" "ps" "th")) ("_3anim123i" "ch" "ch" "ph" "ps" "th" "th" ("id" "ch") "_3animi_1" ("id" "th"))) ;; Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.) (theorem "_3anim3i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3animi_1" ("wi" "ph" "ps"))) (for) ("wi" ("w3a" "ch" "th" "ph") ("w3a" "ch" "th" "ps")) ("_3anim123i" "ch" "ch" "th" "th" "ph" "ps" ("id" "ch") ("id" "th") "_3animi_1")) ;; Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.) (theorem "_3anbi123i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("bi3_1" ("wb" "ph" "ps")) ("bi3_2" ("wb" "ch" "th")) ("bi3_3" ("wb" "ta" "et"))) (for) ("wb" ("w3a" "ph" "ch" "ta") ("w3a" "ps" "th" "et")) ("_3bitr4i" ("wa" ("wa" "ph" "ch") "ta") ("wa" ("wa" "ps" "th") "et") ("w3a" "ph" "ch" "ta") ("w3a" "ps" "th" "et") ("anbi12i" ("wa" "ph" "ch") ("wa" "ps" "th") "ta" "et" ("anbi12i" "ph" "ps" "ch" "th" "bi3_1" "bi3_2") "bi3_3") ("df_3an" "ph" "ch" "ta") ("df_3an" "ps" "th" "et"))) ;; Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.) (theorem "_3orbi123i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("bi3_1" ("wb" "ph" "ps")) ("bi3_2" ("wb" "ch" "th")) ("bi3_3" ("wb" "ta" "et"))) (for) ("wb" ("w3o" "ph" "ch" "ta") ("w3o" "ps" "th" "et")) ("_3bitr4i" ("wo" ("wo" "ph" "ch") "ta") ("wo" ("wo" "ps" "th") "et") ("w3o" "ph" "ch" "ta") ("w3o" "ps" "th" "et") ("orbi12i" ("wo" "ph" "ch") ("wo" "ps" "th") "ta" "et" ("orbi12i" "ph" "ps" "ch" "th" "bi3_1" "bi3_2") "bi3_3") ("df_3or" "ph" "ch" "ta") ("df_3or" "ps" "th" "et"))) ;; Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.) (theorem "_3anbi1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3anbi1i_1" ("wb" "ph" "ps"))) (for) ("wb" ("w3a" "ph" "ch" "th") ("w3a" "ps" "ch" "th")) ("_3anbi123i" "ph" "ps" "ch" "ch" "th" "th" "_3anbi1i_1" ("biid" "ch") ("biid" "th"))) ;; Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.) (theorem "_3anbi2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3anbi1i_1" ("wb" "ph" "ps"))) (for) ("wb" ("w3a" "ch" "ph" "th") ("w3a" "ch" "ps" "th")) ("_3anbi123i" "ch" "ch" "ph" "ps" "th" "th" ("biid" "ch") "_3anbi1i_1" ("biid" "th"))) ;; Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.) (theorem "_3anbi3i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3anbi1i_1" ("wb" "ph" "ps"))) (for) ("wb" ("w3a" "ch" "th" "ph") ("w3a" "ch" "th" "ps")) ("_3anbi123i" "ch" "ch" "th" "th" "ph" "ps" ("biid" "ch") ("biid" "th") "_3anbi1i_1")) ;; Apply ~ ex to a hypothesis with a 3-right-nested conjunction antecedent, with the antecedent of the assertion being a triple conjunction rather than a 2-right-nested conjunction. (Contributed by Alan Sare, 22-Apr-2018.) (theorem "ex3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("ex3_1" ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta"))) (for) ("wi" ("w3a" "ph" "ps" "ch") ("wi" "th" "ta")) ("_3impa" "ph" "ps" "ch" ("wi" "th" "ta") ("ex" ("wa" ("wa" "ph" "ps") "ch") "th" "ta" "ex3_1"))) ;; Obsolete version of ~ 3exp as of 21-Jun-2022. (Contributed by NM, 30-May-1994.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3expOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3comOLD_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))) ("syl8" "ph" "ps" "ch" ("w3a" "ph" "ps" "ch") "th" ("pm3_2an3" "ph" "ps" "ch") "_3comOLD_1")) ;; Obsolete version of ~ 3expa as of 21-Jun-2022. (Contributed by NM, 20-Aug-1995.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3expaOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3comOLD_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" "ph" "ps") "ch") "th") ("imp31" "ph" "ps" "ch" "th" ("_3exp" "ph" "ps" "ch" "th" "_3comOLD_1"))) ;; Obsolete version of ~ 3com12 as of 21-Jun-2022. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3com12OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3comOLD_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ps" "ph" "ch") "th") ("sylbi" ("w3a" "ps" "ph" "ch") ("w3a" "ph" "ps" "ch") "th" ("_3ancoma" "ps" "ph" "ch") "_3comOLD_1")) ;; Obsolete version of ~ 3com13 as of 21-Jun-2022. (Contributed by NM, 28-Jan-1996.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3com13OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3comOLD_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ch" "ps" "ph") "th") ("sylbi" ("w3a" "ch" "ps" "ph") ("w3a" "ph" "ps" "ch") "th" ("_3anrev" "ch" "ps" "ph") "_3comOLD_1")) ;; Obsolete version of ~ 3com23 as of 9-Apr-2022. (Contributed by NM, 28-Jan-1996.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3com23OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3comOLD_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ph" "ch" "ps") "th") ("_3imp" "ph" "ch" "ps" "th" ("com23" "ph" "ps" "ch" "th" ("_3exp" "ph" "ps" "ch" "th" "_3comOLD_1")))) ;; Obsolete version of ~ 3comr as of 9-Apr-2022. (Contributed by NM, 28-Jan-1996.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3comrOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3comOLD_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("w3a" "ch" "ph" "ps") "th") ("_3coml" "ps" "ch" "ph" "th" ("_3coml" "ph" "ps" "ch" "th" "_3comOLD_1"))) ;; Obsolete version of ~ 3imp21 as of 22-Jun-2022. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3imp21OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3imp21OLD_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))) (for) ("wi" ("w3a" "ps" "ph" "ch") "th") ("_3com12" "ph" "ps" "ch" "th" ("_3imp" "ph" "ps" "ch" "th" "_3imp21OLD_1"))) ;; An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 13-Apr-2022.) (theorem "_3imp3i2an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("_3imp3i2an_1" ("wi" ("w3a" "ph" "ps" "ch") "th")) ("_3imp3i2an_2" ("wi" ("wa" "ph" "ch") "ta")) ("_3imp3i2an_3" ("wi" ("wa" "th" "ta") "et"))) (for) ("wi" ("w3a" "ph" "ps" "ch") "et") ("syl2anc" ("w3a" "ph" "ps" "ch") "th" "ta" "et" "_3imp3i2an_1" ("syl" ("w3a" "ph" "ps" "ch") ("wa" "ph" "ch") "ta" ("_3simpb" "ph" "ps" "ch") "_3imp3i2an_2") "_3imp3i2an_3")) ;; Obsolete version of ~ 3imp3i2an as of 13-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3imp3i2anOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("_3imp3i2an_1" ("wi" ("w3a" "ph" "ps" "ch") "th")) ("_3imp3i2an_2" ("wi" ("wa" "ph" "ch") "ta")) ("_3imp3i2an_3" ("wi" ("wa" "th" "ta") "et"))) (for) ("wi" ("w3a" "ph" "ps" "ch") "et") ("_3imp231" "ch" "ph" "ps" "et" ("pm2_43i" "ch" ("wi" "ph" ("wi" "ps" "et")) ("com4r" "ch" "ph" "ps" "ch" "et" ("pm2_43b" "ch" "ph" ("wi" "ps" ("wi" "ch" "et")) ("ex" "ph" "ch" ("wi" "ph" ("wi" "ps" ("wi" "ch" "et"))) ("syl" ("wa" "ph" "ch") "ta" ("wi" "ph" ("wi" "ps" ("wi" "ch" "et"))) "_3imp3i2an_2" ("com4r" "ph" "ps" "ch" "ta" "et" ("syl8" "ph" "ps" "ch" "th" ("wi" "ta" "et") ("_3exp" "ph" "ps" "ch" "th" "_3imp3i2an_1") ("ex" "th" "ta" "et" "_3imp3i2an_3")))))))))) ;; Obsolete version of ~ 3an1rs as of 14-Apr-2022. (Contributed by NM, 16-Dec-2007.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "_3an1rsOLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3an1rsOLD_1" ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta"))) (for) ("wi" ("wa" ("w3a" "ph" "ps" "th") "ch") "ta") ("imp" ("w3a" "ph" "ps" "th") "ch" "ta" ("_3imp" "ph" "ps" "th" ("wi" "ch" "ta") ("com34" "ph" "ps" "ch" "th" "ta" ("_3exp" "ph" "ps" "ch" ("wi" "th" "ta") ("ex" ("w3a" "ph" "ps" "ch") "th" "ta" "_3an1rsOLD_1")))))) ;; Importation to left triple conjunction. (Contributed by NM, 24-Feb-2005.) (theorem "_3imp1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3imp1_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta") ("imp" ("w3a" "ph" "ps" "ch") "th" "ta" ("_3imp" "ph" "ps" "ch" ("wi" "th" "ta") "_3imp1_1"))) ;; Importation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.) (theorem "_3impd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3imp1_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" "ph" ("wi" ("w3a" "ps" "ch" "th") "ta")) ("com12" ("w3a" "ps" "ch" "th") "ph" "ta" ("_3imp" "ps" "ch" "th" ("wi" "ph" "ta") ("com4l" "ph" "ps" "ch" "th" "ta" "_3imp1_1")))) ;; Importation to right triple conjunction. (Contributed by NM, 26-Oct-2006.) (theorem "_3imp2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3imp1_1" ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))))) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ta") ("imp" "ph" ("w3a" "ps" "ch" "th") "ta" ("_3impd" "ph" "ps" "ch" "th" "ta" "_3imp1_1"))) ;; Exportation from left triple conjunction. (Contributed by NM, 24-Feb-2005.) (theorem "_3exp1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3exp1_1" ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("_3exp" "ph" "ps" "ch" ("wi" "th" "ta") ("ex" ("w3a" "ph" "ps" "ch") "th" "ta" "_3exp1_1"))) ;; Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.) (theorem "_3expd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3expd_1" ("wi" "ph" ("wi" ("w3a" "ps" "ch" "th") "ta")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("com4r" "ps" "ch" "th" "ph" "ta" ("_3exp" "ps" "ch" "th" ("wi" "ph" "ta") ("com12" "ph" ("w3a" "ps" "ch" "th") "ta" "_3expd_1")))) ;; Exportation from right triple conjunction. (Contributed by NM, 26-Oct-2006.) (theorem "_3exp2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3exp2_1" ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ta"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta")))) ("_3expd" "ph" "ps" "ch" "th" "ta" ("ex" "ph" ("w3a" "ps" "ch" "th") "ta" "_3exp2_1"))) ;; A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.) (theorem "exp5o" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("exp5o_1" ("wi" ("w3a" "ph" "ps" "ch") ("wi" ("wa" "th" "ta") "et")))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))) ("_3exp" "ph" "ps" "ch" ("wi" "th" ("wi" "ta" "et")) ("expd" ("w3a" "ph" "ps" "ch") "th" "ta" "et" "exp5o_1"))) ;; A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.) (theorem "exp516" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("exp516_1" ("wi" ("wa" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ta") "et"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))) ("_3expd" "ph" "ps" "ch" "th" ("wi" "ta" "et") ("exp31" "ph" ("w3a" "ps" "ch" "th") "ta" "et" "exp516_1"))) ;; A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.) (theorem "exp520" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("exp520_1" ("wi" ("wa" ("w3a" "ph" "ps" "ch") ("wa" "th" "ta")) "et"))) (for) ("wi" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" ("wi" "ta" "et"))))) ("exp5o" "ph" "ps" "ch" "th" "ta" "et" ("ex" ("w3a" "ph" "ps" "ch") ("wa" "th" "ta") "et" "exp520_1"))) ;; Version of ~ impexp for a triple conjunction. (Contributed by Alan Sare, 31-Dec-2011.) (theorem "_3impexp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wi" ("w3a" "ph" "ps" "ch") "th") ("wi" "ph" ("wi" "ps" ("wi" "ch" "th")))) ("impbii" ("wi" ("w3a" "ph" "ps" "ch") "th") ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))) ("_3expd" ("wi" ("w3a" "ph" "ps" "ch") "th") "ph" "ps" "ch" "th" ("id" ("wi" ("w3a" "ph" "ps" "ch") "th"))) ("_3impd" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))) "ph" "ps" "ch" "th" ("id" ("wi" "ph" ("wi" "ps" ("wi" "ch" "th"))))))) ;; Swap conjuncts. (Contributed by NM, 16-Dec-2007.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "_3an1rs" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3an1rs_1" ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta"))) (for) ("wi" ("wa" ("w3a" "ph" "ps" "th") "ch") "ta") ("_3imp1" "ph" "ps" "th" "ch" "ta" ("com34" "ph" "ps" "ch" "th" "ta" ("_3exp1" "ph" "ps" "ch" "th" "ta" "_3an1rs_1")))) ;; Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.) (theorem "_3anassrs" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3anassrs_1" ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ta"))) (for) ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta") ("imp41" "ph" "ps" "ch" "th" "ta" ("_3exp2" "ph" "ps" "ch" "th" "ta" "_3anassrs_1"))) ;; Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (theorem "_3an4anass" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wa" ("w3a" "ph" "ps" "ch") "th") ("wa" ("wa" "ph" "ps") ("wa" "ch" "th"))) ("bitri" ("wa" ("w3a" "ph" "ps" "ch") "th") ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") ("wa" ("wa" "ph" "ps") ("wa" "ch" "th")) ("anbi1i" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") "ch") "th" ("df_3an" "ph" "ps" "ch")) ("anass" ("wa" "ph" "ps") "ch" "th"))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad5ant245" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ta" "ph") "et") "ps") "ch") "th") ("ad4ant134" ("wa" "ta" "ph") "ps" "ch" "th" "et" ("_3adant1l" "ph" "ps" "ch" "th" "ta" "ad5ant_1"))) ;; Obsolete version of ~ ad5ant245 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant245OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ta" "ph") "et") "ps") "ch") "th") ("imp41" ("wa" "ta" "ph") "et" "ps" "ch" "th" ("imp" "ta" "ph" ("wi" "et" ("wi" "ps" ("wi" "ch" "th"))) ("a1i13" "ta" "ph" "et" ("wi" "ps" ("wi" "ch" "th")) ("_3exp" "ph" "ps" "ch" "th" "ad5ant_1"))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad5ant234" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ta" "ph") "ps") "ch") "et") "th") ("adantr" ("wa" ("wa" ("wa" "ta" "ph") "ps") "ch") "th" "et" ("ad4ant234" "ph" "ps" "ch" "th" "ta" "ad5ant_1"))) ;; Obsolete version of ~ ad5ant234 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant234OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ta" "ph") "ps") "ch") "et") "th") ("imp41" ("wa" "ta" "ph") "ps" "ch" "et" "th" ("imp" "ta" "ph" ("wi" "ps" ("wi" "ch" ("wi" "et" "th"))) ("com5r" "ph" "ps" "ch" "et" "ta" "th" ("a1ddd" "ph" "ps" "ch" "et" ("wi" "ta" "th") ("a1ddd" "ph" "ps" "ch" "ta" "th" ("_3exp" "ph" "ps" "ch" "th" "ad5ant_1"))))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) (theorem "ad5ant235" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ta" "ph") "ps") "et") "ch") "th") ("adantlr" ("wa" ("wa" "ta" "ph") "ps") "ch" "th" "et" ("ad4ant234" "ph" "ps" "ch" "th" "ta" "ad5ant_1"))) ;; Obsolete version of ~ ad5ant235 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant235OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ta" "ph") "ps") "et") "ch") "th") ("imp41" ("wa" "ta" "ph") "ps" "et" "ch" "th" ("imp" "ta" "ph" ("wi" "ps" ("wi" "et" ("wi" "ch" "th"))) ("com45" "ta" "ph" "ps" "ch" "et" "th" ("com5r" "ph" "ps" "ch" "et" "ta" "th" ("a1ddd" "ph" "ps" "ch" "et" ("wi" "ta" "th") ("a1ddd" "ph" "ps" "ch" "ta" "th" ("_3exp" "ph" "ps" "ch" "th" "ad5ant_1")))))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "ad5ant123" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "ta") "et") "th") ("ad2antrr" ("wa" ("wa" "ph" "ps") "ch") "th" "ta" "et" ("_3expa" "ph" "ps" "ch" "th" "ad5ant_1"))) ;; Obsolete version of ~ ad5ant123 as of 23-Jun-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant123OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ch") "ta") "et") "th") ("imp41" ("wa" "ph" "ps") "ch" "ta" "et" "th" ("imp" "ph" "ps" ("wi" "ch" ("wi" "ta" ("wi" "et" "th"))) ("com45" "ph" "ps" "ch" "et" "ta" "th" ("a1ddd" "ph" "ps" "ch" "et" ("wi" "ta" "th") ("a1ddd" "ph" "ps" "ch" "ta" "th" ("_3exp" "ph" "ps" "ch" "th" "ad5ant_1"))))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "ad5ant124" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ta") "ch") "et") "th") ("adantr" ("wa" ("wa" ("wa" "ph" "ps") "ta") "ch") "th" "et" ("ad4ant124" "ph" "ps" "ch" "th" "ta" "ad5ant_1"))) ;; Obsolete version of ~ ad5ant124 as of 23-Jun-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant124OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ta") "ch") "et") "th") ("imp41" ("wa" "ph" "ps") "ta" "ch" "et" "th" ("imp" "ph" "ps" ("wi" "ta" ("wi" "ch" ("wi" "et" "th"))) ("com34" "ph" "ps" "ch" "ta" ("wi" "et" "th") ("com45" "ph" "ps" "ch" "et" "ta" "th" ("a1ddd" "ph" "ps" "ch" "et" ("wi" "ta" "th") ("a1ddd" "ph" "ps" "ch" "ta" "th" ("_3exp" "ph" "ps" "ch" "th" "ad5ant_1")))))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "ad5ant125" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ta") "et") "ch") "th") ("imp41" ("wa" "ph" "ps") "ta" "et" "ch" "th" ("_2a1d" ("wa" "ph" "ps") ("wi" "ch" "th") "ta" "et" ("_3expia" "ph" "ps" "ch" "th" "ad5ant_1")))) ;; Obsolete version of ~ ad5ant125 as of 23-Jun-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant125OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ps") "ta") "et") "ch") "th") ("imp41" ("wa" "ph" "ps") "ta" "et" "ch" "th" ("imp" "ph" "ps" ("wi" "ta" ("wi" "et" ("wi" "ch" "th"))) ("_2a1dd" "ph" "ps" ("wi" "ch" "th") "ta" "et" ("_3exp" "ph" "ps" "ch" "th" "ad5ant_1"))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "ad5ant134" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ta") "ps") "ch") "et") "th") ("adantr" ("wa" ("wa" ("wa" "ph" "ta") "ps") "ch") "th" "et" ("ad4ant134" "ph" "ps" "ch" "th" "ta" "ad5ant_1"))) ;; Obsolete version of ~ ad5ant134 as of 23-Jun-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant134OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ta") "ps") "ch") "et") "th") ("imp41" ("wa" "ph" "ta") "ps" "ch" "et" "th" ("imp" "ph" "ta" ("wi" "ps" ("wi" "ch" ("wi" "et" "th"))) ("com23" "ph" "ps" "ta" ("wi" "ch" ("wi" "et" "th")) ("com34" "ph" "ps" "ch" "ta" ("wi" "et" "th") ("com45" "ph" "ps" "ch" "et" "ta" "th" ("a1ddd" "ph" "ps" "ch" "et" ("wi" "ta" "th") ("a1ddd" "ph" "ps" "ch" "ta" "th" ("_3exp" "ph" "ps" "ch" "th" "ad5ant_1"))))))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "ad5ant135" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ta") "ps") "et") "ch") "th") ("adantlr" ("wa" ("wa" "ph" "ta") "ps") "ch" "th" "et" ("ad4ant134" "ph" "ps" "ch" "th" "ta" "ad5ant_1"))) ;; Obsolete version of ~ ad5ant135 as of 23-Jun-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant135OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ta") "ps") "et") "ch") "th") ("imp41" ("wa" "ph" "ta") "ps" "et" "ch" "th" ("imp" "ph" "ta" ("wi" "ps" ("wi" "et" ("wi" "ch" "th"))) ("com45" "ph" "ta" "ps" "ch" "et" "th" ("com23" "ph" "ps" "ta" ("wi" "ch" ("wi" "et" "th")) ("com34" "ph" "ps" "ch" "ta" ("wi" "et" "th") ("com45" "ph" "ps" "ch" "et" "ta" "th" ("a1ddd" "ph" "ps" "ch" "et" ("wi" "ta" "th") ("a1ddd" "ph" "ps" "ch" "ta" "th" ("_3exp" "ph" "ps" "ch" "th" "ad5ant_1")))))))))) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (theorem "ad5ant145" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ta") "et") "ps") "ch") "th") ("adantllr" ("wa" "ph" "ta") "ps" "ch" "th" "et" ("ad4ant134" "ph" "ps" "ch" "th" "ta" "ad5ant_1"))) ;; Obsolete version of ~ ad5ant145 as of 23-Jun-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant145OLD" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant_1" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "ta") "et") "ps") "ch") "th") ("imp41" ("wa" "ph" "ta") "et" "ps" "ch" "th" ("imp" "ph" "ta" ("wi" "et" ("wi" "ps" ("wi" "ch" "th"))) ("_2a1d" "ph" ("wi" "ps" ("wi" "ch" "th")) "ta" "et" ("_3exp" "ph" "ps" "ch" "th" "ad5ant_1"))))) ;; Obsolete as of 17-May-2022. Use ~ adantl3r instead. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "ad5ant1345" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant1345_1" ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "ph" "et") "ps") "ch") "th") "ta") ("adantl3r" "ph" "ps" "ch" "th" "ta" "et" "ad5ant1345_1")) ;; Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (theorem "ad5ant2345" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("ad5ant2345_1" ("wi" ("wa" ("wa" ("wa" "ph" "ps") "ch") "th") "ta"))) (for) ("wi" ("wa" ("wa" ("wa" ("wa" "et" "ph") "ps") "ch") "th") "ta") ("imp41" ("wa" "et" "ph") "ps" "ch" "th" "ta" ("adantl" "ph" ("wi" "ps" ("wi" "ch" ("wi" "th" "ta"))) "et" ("exp41" "ph" "ps" "ch" "th" "ta" "ad5ant2345_1")))) ;; Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.) (theorem "syl12anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl12anc_4" ("wi" ("wa" "ps" ("wa" "ch" "th")) "ta"))) (for) ("wi" "ph" "ta") ("syl" "ph" ("wa" "ps" ("wa" "ch" "th")) "ta" ("jca32" "ph" "ps" "ch" "th" "syl12anc_1" "syl12anc_2" "syl12anc_3") "syl12anc_4")) ;; Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.) (theorem "syl21anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl21anc_4" ("wi" ("wa" ("wa" "ps" "ch") "th") "ta"))) (for) ("wi" "ph" "ta") ("syl" "ph" ("wa" ("wa" "ps" "ch") "th") "ta" ("jca31" "ph" "ps" "ch" "th" "syl12anc_1" "syl12anc_2" "syl12anc_3") "syl21anc_4")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl3anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl3anc_4" ("wi" ("w3a" "ps" "ch" "th") "ta"))) (for) ("wi" "ph" "ta") ("syl" "ph" ("w3a" "ps" "ch" "th") "ta" ("_3jca" "ph" "ps" "ch" "th" "syl12anc_1" "syl12anc_2" "syl12anc_3") "syl3anc_4")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl22anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl22anc_5" ("wi" ("wa" ("wa" "ps" "ch") ("wa" "th" "ta")) "et"))) (for) ("wi" "ph" "et") ("syl12anc" "ph" ("wa" "ps" "ch") "th" "ta" "et" ("jca" "ph" "ps" "ch" "syl12anc_1" "syl12anc_2") "syl12anc_3" "syl22anc_4" "syl22anc_5")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl13anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl13anc_5" ("wi" ("wa" "ps" ("w3a" "ch" "th" "ta")) "et"))) (for) ("wi" "ph" "et") ("syl2anc" "ph" "ps" ("w3a" "ch" "th" "ta") "et" "syl12anc_1" ("_3jca" "ph" "ch" "th" "ta" "syl12anc_2" "syl12anc_3" "syl22anc_4") "syl13anc_5")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl31anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl31anc_5" ("wi" ("wa" ("w3a" "ps" "ch" "th") "ta") "et"))) (for) ("wi" "ph" "et") ("syl2anc" "ph" ("w3a" "ps" "ch" "th") "ta" "et" ("_3jca" "ph" "ps" "ch" "th" "syl12anc_1" "syl12anc_2" "syl12anc_3") "syl22anc_4" "syl31anc_5")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl112anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl112anc_5" ("wi" ("w3a" "ps" "ch" ("wa" "th" "ta")) "et"))) (for) ("wi" "ph" "et") ("syl3anc" "ph" "ps" "ch" ("wa" "th" "ta") "et" "syl12anc_1" "syl12anc_2" ("jca" "ph" "th" "ta" "syl12anc_3" "syl22anc_4") "syl112anc_5")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl121anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl121anc_5" ("wi" ("w3a" "ps" ("wa" "ch" "th") "ta") "et"))) (for) ("wi" "ph" "et") ("syl3anc" "ph" "ps" ("wa" "ch" "th") "ta" "et" "syl12anc_1" ("jca" "ph" "ch" "th" "syl12anc_2" "syl12anc_3") "syl22anc_4" "syl121anc_5")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl211anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl211anc_5" ("wi" ("w3a" ("wa" "ps" "ch") "th" "ta") "et"))) (for) ("wi" "ph" "et") ("syl3anc" "ph" ("wa" "ps" "ch") "th" "ta" "et" ("jca" "ph" "ps" "ch" "syl12anc_1" "syl12anc_2") "syl12anc_3" "syl22anc_4" "syl211anc_5")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl23anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl23anc_6" ("wi" ("wa" ("wa" "ps" "ch") ("w3a" "th" "ta" "et")) "ze"))) (for) ("wi" "ph" "ze") ("syl13anc" "ph" ("wa" "ps" "ch") "th" "ta" "et" "ze" ("jca" "ph" "ps" "ch" "syl12anc_1" "syl12anc_2") "syl12anc_3" "syl22anc_4" "syl23anc_5" "syl23anc_6")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl32anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl32anc_6" ("wi" ("wa" ("w3a" "ps" "ch" "th") ("wa" "ta" "et")) "ze"))) (for) ("wi" "ph" "ze") ("syl31anc" "ph" "ps" "ch" "th" ("wa" "ta" "et") "ze" "syl12anc_1" "syl12anc_2" "syl12anc_3" ("jca" "ph" "ta" "et" "syl22anc_4" "syl23anc_5") "syl32anc_6")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl122anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl122anc_6" ("wi" ("w3a" "ps" ("wa" "ch" "th") ("wa" "ta" "et")) "ze"))) (for) ("wi" "ph" "ze") ("syl121anc" "ph" "ps" "ch" "th" ("wa" "ta" "et") "ze" "syl12anc_1" "syl12anc_2" "syl12anc_3" ("jca" "ph" "ta" "et" "syl22anc_4" "syl23anc_5") "syl122anc_6")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl212anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl212anc_6" ("wi" ("w3a" ("wa" "ps" "ch") "th" ("wa" "ta" "et")) "ze"))) (for) ("wi" "ph" "ze") ("syl211anc" "ph" "ps" "ch" "th" ("wa" "ta" "et") "ze" "syl12anc_1" "syl12anc_2" "syl12anc_3" ("jca" "ph" "ta" "et" "syl22anc_4" "syl23anc_5") "syl212anc_6")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl221anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl221anc_6" ("wi" ("w3a" ("wa" "ps" "ch") ("wa" "th" "ta") "et") "ze"))) (for) ("wi" "ph" "ze") ("syl211anc" "ph" "ps" "ch" ("wa" "th" "ta") "et" "ze" "syl12anc_1" "syl12anc_2" ("jca" "ph" "th" "ta" "syl12anc_3" "syl22anc_4") "syl23anc_5" "syl221anc_6")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl113anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl113anc_6" ("wi" ("w3a" "ps" "ch" ("w3a" "th" "ta" "et")) "ze"))) (for) ("wi" "ph" "ze") ("syl3anc" "ph" "ps" "ch" ("w3a" "th" "ta" "et") "ze" "syl12anc_1" "syl12anc_2" ("_3jca" "ph" "th" "ta" "et" "syl12anc_3" "syl22anc_4" "syl23anc_5") "syl113anc_6")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl131anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl131anc_6" ("wi" ("w3a" "ps" ("w3a" "ch" "th" "ta") "et") "ze"))) (for) ("wi" "ph" "ze") ("syl3anc" "ph" "ps" ("w3a" "ch" "th" "ta") "et" "ze" "syl12anc_1" ("_3jca" "ph" "ch" "th" "ta" "syl12anc_2" "syl12anc_3" "syl22anc_4") "syl23anc_5" "syl131anc_6")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl311anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl311anc_6" ("wi" ("w3a" ("w3a" "ps" "ch" "th") "ta" "et") "ze"))) (for) ("wi" "ph" "ze") ("syl3anc" "ph" ("w3a" "ps" "ch" "th") "ta" "et" "ze" ("_3jca" "ph" "ps" "ch" "th" "syl12anc_1" "syl12anc_2" "syl12anc_3") "syl22anc_4" "syl23anc_5" "syl311anc_6")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl33anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl33anc_7" ("wi" ("wa" ("w3a" "ps" "ch" "th") ("w3a" "ta" "et" "ze")) "si"))) (for) ("wi" "ph" "si") ("syl13anc" "ph" ("w3a" "ps" "ch" "th") "ta" "et" "ze" "si" ("_3jca" "ph" "ps" "ch" "th" "syl12anc_1" "syl12anc_2" "syl12anc_3") "syl22anc_4" "syl23anc_5" "syl33anc_6" "syl33anc_7")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl222anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl222anc_7" ("wi" ("w3a" ("wa" "ps" "ch") ("wa" "th" "ta") ("wa" "et" "ze")) "si"))) (for) ("wi" "ph" "si") ("syl221anc" "ph" "ps" "ch" "th" "ta" ("wa" "et" "ze") "si" "syl12anc_1" "syl12anc_2" "syl12anc_3" "syl22anc_4" ("jca" "ph" "et" "ze" "syl23anc_5" "syl33anc_6") "syl222anc_7")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl123anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl123anc_7" ("wi" ("w3a" "ps" ("wa" "ch" "th") ("w3a" "ta" "et" "ze")) "si"))) (for) ("wi" "ph" "si") ("syl113anc" "ph" "ps" ("wa" "ch" "th") "ta" "et" "ze" "si" "syl12anc_1" ("jca" "ph" "ch" "th" "syl12anc_2" "syl12anc_3") "syl22anc_4" "syl23anc_5" "syl33anc_6" "syl123anc_7")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.) (theorem "syl132anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl132anc_7" ("wi" ("w3a" "ps" ("w3a" "ch" "th" "ta") ("wa" "et" "ze")) "si"))) (for) ("wi" "ph" "si") ("syl131anc" "ph" "ps" "ch" "th" "ta" ("wa" "et" "ze") "si" "syl12anc_1" "syl12anc_2" "syl12anc_3" "syl22anc_4" ("jca" "ph" "et" "ze" "syl23anc_5" "syl33anc_6") "syl132anc_7")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl213anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl213anc_7" ("wi" ("w3a" ("wa" "ps" "ch") "th" ("w3a" "ta" "et" "ze")) "si"))) (for) ("wi" "ph" "si") ("syl113anc" "ph" ("wa" "ps" "ch") "th" "ta" "et" "ze" "si" ("jca" "ph" "ps" "ch" "syl12anc_1" "syl12anc_2") "syl12anc_3" "syl22anc_4" "syl23anc_5" "syl33anc_6" "syl213anc_7")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl231anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl231anc_7" ("wi" ("w3a" ("wa" "ps" "ch") ("w3a" "th" "ta" "et") "ze") "si"))) (for) ("wi" "ph" "si") ("syl131anc" "ph" ("wa" "ps" "ch") "th" "ta" "et" "ze" "si" ("jca" "ph" "ps" "ch" "syl12anc_1" "syl12anc_2") "syl12anc_3" "syl22anc_4" "syl23anc_5" "syl33anc_6" "syl231anc_7")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.) (theorem "syl312anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl312anc_7" ("wi" ("w3a" ("w3a" "ps" "ch" "th") "ta" ("wa" "et" "ze")) "si"))) (for) ("wi" "ph" "si") ("syl311anc" "ph" "ps" "ch" "th" "ta" ("wa" "et" "ze") "si" "syl12anc_1" "syl12anc_2" "syl12anc_3" "syl22anc_4" ("jca" "ph" "et" "ze" "syl23anc_5" "syl33anc_6") "syl312anc_7")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.) (theorem "syl321anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl321anc_7" ("wi" ("w3a" ("w3a" "ps" "ch" "th") ("wa" "ta" "et") "ze") "si"))) (for) ("wi" "ph" "si") ("syl311anc" "ph" "ps" "ch" "th" ("wa" "ta" "et") "ze" "si" "syl12anc_1" "syl12anc_2" "syl12anc_3" ("jca" "ph" "ta" "et" "syl22anc_4" "syl23anc_5") "syl33anc_6" "syl321anc_7")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl133anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl133anc_7" ("wi" "ph" "si")) ("syl133anc_8" ("wi" ("w3a" "ps" ("w3a" "ch" "th" "ta") ("w3a" "et" "ze" "si")) "rh"))) (for) ("wi" "ph" "rh") ("syl131anc" "ph" "ps" "ch" "th" "ta" ("w3a" "et" "ze" "si") "rh" "syl12anc_1" "syl12anc_2" "syl12anc_3" "syl22anc_4" ("_3jca" "ph" "et" "ze" "si" "syl23anc_5" "syl33anc_6" "syl133anc_7") "syl133anc_8")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl313anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl133anc_7" ("wi" "ph" "si")) ("syl313anc_8" ("wi" ("w3a" ("w3a" "ps" "ch" "th") "ta" ("w3a" "et" "ze" "si")) "rh"))) (for) ("wi" "ph" "rh") ("syl311anc" "ph" "ps" "ch" "th" "ta" ("w3a" "et" "ze" "si") "rh" "syl12anc_1" "syl12anc_2" "syl12anc_3" "syl22anc_4" ("_3jca" "ph" "et" "ze" "si" "syl23anc_5" "syl33anc_6" "syl133anc_7") "syl313anc_8")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl331anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl133anc_7" ("wi" "ph" "si")) ("syl331anc_8" ("wi" ("w3a" ("w3a" "ps" "ch" "th") ("w3a" "ta" "et" "ze") "si") "rh"))) (for) ("wi" "ph" "rh") ("syl311anc" "ph" "ps" "ch" "th" ("w3a" "ta" "et" "ze") "si" "rh" "syl12anc_1" "syl12anc_2" "syl12anc_3" ("_3jca" "ph" "ta" "et" "ze" "syl22anc_4" "syl23anc_5" "syl33anc_6") "syl133anc_7" "syl331anc_8")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl223anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl133anc_7" ("wi" "ph" "si")) ("syl223anc_8" ("wi" ("w3a" ("wa" "ps" "ch") ("wa" "th" "ta") ("w3a" "et" "ze" "si")) "rh"))) (for) ("wi" "ph" "rh") ("syl213anc" "ph" "ps" "ch" ("wa" "th" "ta") "et" "ze" "si" "rh" "syl12anc_1" "syl12anc_2" ("jca" "ph" "th" "ta" "syl12anc_3" "syl22anc_4") "syl23anc_5" "syl33anc_6" "syl133anc_7" "syl223anc_8")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl232anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl133anc_7" ("wi" "ph" "si")) ("syl232anc_8" ("wi" ("w3a" ("wa" "ps" "ch") ("w3a" "th" "ta" "et") ("wa" "ze" "si")) "rh"))) (for) ("wi" "ph" "rh") ("syl231anc" "ph" "ps" "ch" "th" "ta" "et" ("wa" "ze" "si") "rh" "syl12anc_1" "syl12anc_2" "syl12anc_3" "syl22anc_4" "syl23anc_5" ("jca" "ph" "ze" "si" "syl33anc_6" "syl133anc_7") "syl232anc_8")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl322anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl133anc_7" ("wi" "ph" "si")) ("syl322anc_8" ("wi" ("w3a" ("w3a" "ps" "ch" "th") ("wa" "ta" "et") ("wa" "ze" "si")) "rh"))) (for) ("wi" "ph" "rh") ("syl321anc" "ph" "ps" "ch" "th" "ta" "et" ("wa" "ze" "si") "rh" "syl12anc_1" "syl12anc_2" "syl12anc_3" "syl22anc_4" "syl23anc_5" ("jca" "ph" "ze" "si" "syl33anc_6" "syl133anc_7") "syl322anc_8")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl233anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl133anc_7" ("wi" "ph" "si")) ("syl233anc_8" ("wi" "ph" "rh")) ("syl233anc_9" ("wi" ("w3a" ("wa" "ps" "ch") ("w3a" "th" "ta" "et") ("w3a" "ze" "si" "rh")) "mu"))) (for) ("wi" "ph" "mu") ("syl133anc" "ph" ("wa" "ps" "ch") "th" "ta" "et" "ze" "si" "rh" "mu" ("jca" "ph" "ps" "ch" "syl12anc_1" "syl12anc_2") "syl12anc_3" "syl22anc_4" "syl23anc_5" "syl33anc_6" "syl133anc_7" "syl233anc_8" "syl233anc_9")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl323anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl133anc_7" ("wi" "ph" "si")) ("syl233anc_8" ("wi" "ph" "rh")) ("syl323anc_9" ("wi" ("w3a" ("w3a" "ps" "ch" "th") ("wa" "ta" "et") ("w3a" "ze" "si" "rh")) "mu"))) (for) ("wi" "ph" "mu") ("syl313anc" "ph" "ps" "ch" "th" ("wa" "ta" "et") "ze" "si" "rh" "mu" "syl12anc_1" "syl12anc_2" "syl12anc_3" ("jca" "ph" "ta" "et" "syl22anc_4" "syl23anc_5") "syl33anc_6" "syl133anc_7" "syl233anc_8" "syl323anc_9")) ;; Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) (theorem "syl332anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl133anc_7" ("wi" "ph" "si")) ("syl233anc_8" ("wi" "ph" "rh")) ("syl332anc_9" ("wi" ("w3a" ("w3a" "ps" "ch" "th") ("w3a" "ta" "et" "ze") ("wa" "si" "rh")) "mu"))) (for) ("wi" "ph" "mu") ("syl331anc" "ph" "ps" "ch" "th" "ta" "et" "ze" ("wa" "si" "rh") "mu" "syl12anc_1" "syl12anc_2" "syl12anc_3" "syl22anc_4" "syl23anc_5" "syl33anc_6" ("jca" "ph" "si" "rh" "syl133anc_7" "syl233anc_8") "syl332anc_9")) ;; A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.) (theorem "syl333anc" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff")) ("mu" ( "wff")) ("la" ( "wff"))) (for ("syl12anc_1" ("wi" "ph" "ps")) ("syl12anc_2" ("wi" "ph" "ch")) ("syl12anc_3" ("wi" "ph" "th")) ("syl22anc_4" ("wi" "ph" "ta")) ("syl23anc_5" ("wi" "ph" "et")) ("syl33anc_6" ("wi" "ph" "ze")) ("syl133anc_7" ("wi" "ph" "si")) ("syl233anc_8" ("wi" "ph" "rh")) ("syl333anc_9" ("wi" "ph" "mu")) ("syl333anc_10" ("wi" ("w3a" ("w3a" "ps" "ch" "th") ("w3a" "ta" "et" "ze") ("w3a" "si" "rh" "mu")) "la"))) (for) ("wi" "ph" "la") ("syl331anc" "ph" "ps" "ch" "th" "ta" "et" "ze" ("w3a" "si" "rh" "mu") "la" "syl12anc_1" "syl12anc_2" "syl12anc_3" "syl22anc_4" "syl23anc_5" "syl33anc_6" ("_3jca" "ph" "si" "rh" "mu" "syl133anc_7" "syl233anc_8" "syl333anc_9") "syl333anc_10")) ;; A syllogism inference. (Contributed by NM, 22-Aug-1995.) (theorem "syl3an1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl3an1_1" ("wi" "ph" "ps")) ("syl3an1_2" ("wi" ("w3a" "ps" "ch" "th") "ta"))) (for) ("wi" ("w3a" "ph" "ch" "th") "ta") ("syl" ("w3a" "ph" "ch" "th") ("w3a" "ps" "ch" "th") "ta" ("_3anim1i" "ph" "ps" "ch" "th" "syl3an1_1") "syl3an1_2")) ;; A syllogism inference. (Contributed by NM, 22-Aug-1995.) (theorem "syl3an2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl3an2_1" ("wi" "ph" "ch")) ("syl3an2_2" ("wi" ("w3a" "ps" "ch" "th") "ta"))) (for) ("wi" ("w3a" "ps" "ph" "th") "ta") ("_3imp" "ps" "ph" "th" "ta" ("syl5" "ph" "ch" "ps" ("wi" "th" "ta") "syl3an2_1" ("_3exp" "ps" "ch" "th" "ta" "syl3an2_2")))) ;; A syllogism inference. (Contributed by NM, 22-Aug-1995.) (theorem "syl3an3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl3an3_1" ("wi" "ph" "th")) ("syl3an3_2" ("wi" ("w3a" "ps" "ch" "th") "ta"))) (for) ("wi" ("w3a" "ps" "ch" "ph") "ta") ("_3imp" "ps" "ch" "ph" "ta" ("syl7" "ph" "th" "ps" "ch" "ta" "syl3an3_1" ("_3exp" "ps" "ch" "th" "ta" "syl3an3_2")))) ;; A syllogism inference. (Contributed by NM, 22-Aug-1995.) (theorem "syl3an1b" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl3an1b_1" ("wb" "ph" "ps")) ("syl3an1b_2" ("wi" ("w3a" "ps" "ch" "th") "ta"))) (for) ("wi" ("w3a" "ph" "ch" "th") "ta") ("syl3an1" "ph" "ps" "ch" "th" "ta" ("biimpi" "ph" "ps" "syl3an1b_1") "syl3an1b_2")) ;; A syllogism inference. (Contributed by NM, 22-Aug-1995.) (theorem "syl3an2b" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl3an2b_1" ("wb" "ph" "ch")) ("syl3an2b_2" ("wi" ("w3a" "ps" "ch" "th") "ta"))) (for) ("wi" ("w3a" "ps" "ph" "th") "ta") ("syl3an2" "ph" "ps" "ch" "th" "ta" ("biimpi" "ph" "ch" "syl3an2b_1") "syl3an2b_2")) ;; A syllogism inference. (Contributed by NM, 22-Aug-1995.) (theorem "syl3an3b" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl3an3b_1" ("wb" "ph" "th")) ("syl3an3b_2" ("wi" ("w3a" "ps" "ch" "th") "ta"))) (for) ("wi" ("w3a" "ps" "ch" "ph") "ta") ("syl3an3" "ph" "ps" "ch" "th" "ta" ("biimpi" "ph" "th" "syl3an3b_1") "syl3an3b_2")) ;; A syllogism inference. (Contributed by NM, 22-Aug-1995.) (theorem "syl3an1br" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl3an1br_1" ("wb" "ps" "ph")) ("syl3an1br_2" ("wi" ("w3a" "ps" "ch" "th") "ta"))) (for) ("wi" ("w3a" "ph" "ch" "th") "ta") ("syl3an1" "ph" "ps" "ch" "th" "ta" ("biimpri" "ps" "ph" "syl3an1br_1") "syl3an1br_2")) ;; A syllogism inference. (Contributed by NM, 22-Aug-1995.) (theorem "syl3an2br" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl3an2br_1" ("wb" "ch" "ph")) ("syl3an2br_2" ("wi" ("w3a" "ps" "ch" "th") "ta"))) (for) ("wi" ("w3a" "ps" "ph" "th") "ta") ("syl3an2" "ph" "ps" "ch" "th" "ta" ("biimpri" "ch" "ph" "syl3an2br_1") "syl3an2br_2")) ;; A syllogism inference. (Contributed by NM, 22-Aug-1995.) (theorem "syl3an3br" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syl3an3br_1" ("wb" "th" "ph")) ("syl3an3br_2" ("wi" ("w3a" "ps" "ch" "th") "ta"))) (for) ("wi" ("w3a" "ps" "ch" "ph") "ta") ("syl3an3" "ph" "ps" "ch" "th" "ta" ("biimpri" "th" "ph" "syl3an3br_1") "syl3an3br_2")) ;; A triple syllogism inference. (Contributed by NM, 13-May-2004.) (theorem "syl3an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl3an_1" ("wi" "ph" "ps")) ("syl3an_2" ("wi" "ch" "th")) ("syl3an_3" ("wi" "ta" "et")) ("syl3an_4" ("wi" ("w3a" "ps" "th" "et") "ze"))) (for) ("wi" ("w3a" "ph" "ch" "ta") "ze") ("syl" ("w3a" "ph" "ch" "ta") ("w3a" "ps" "th" "et") "ze" ("_3anim123i" "ph" "ps" "ch" "th" "ta" "et" "syl3an_1" "syl3an_2" "syl3an_3") "syl3an_4")) ;; A triple syllogism inference. (Contributed by NM, 15-Oct-2005.) (theorem "syl3anb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl3anb_1" ("wb" "ph" "ps")) ("syl3anb_2" ("wb" "ch" "th")) ("syl3anb_3" ("wb" "ta" "et")) ("syl3anb_4" ("wi" ("w3a" "ps" "th" "et") "ze"))) (for) ("wi" ("w3a" "ph" "ch" "ta") "ze") ("sylbi" ("w3a" "ph" "ch" "ta") ("w3a" "ps" "th" "et") "ze" ("_3anbi123i" "ph" "ps" "ch" "th" "ta" "et" "syl3anb_1" "syl3anb_2" "syl3anb_3") "syl3anb_4")) ;; A triple syllogism inference. (Contributed by NM, 29-Dec-2011.) (theorem "syl3anbr" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl3anbr_1" ("wb" "ps" "ph")) ("syl3anbr_2" ("wb" "th" "ch")) ("syl3anbr_3" ("wb" "et" "ta")) ("syl3anbr_4" ("wi" ("w3a" "ps" "th" "et") "ze"))) (for) ("wi" ("w3a" "ph" "ch" "ta") "ze") ("syl3anb" "ph" "ps" "ch" "th" "ta" "et" "ze" ("bicomi" "ps" "ph" "syl3anbr_1") ("bicomi" "th" "ch" "syl3anbr_2") ("bicomi" "et" "ta" "syl3anbr_3") "syl3anbr_4")) ;; A syllogism inference. (Contributed by NM, 20-May-2007.) (theorem "syld3an3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syld3an3_1" ("wi" ("w3a" "ph" "ps" "ch") "th")) ("syld3an3_2" ("wi" ("w3a" "ph" "ps" "th") "ta"))) (for) ("wi" ("w3a" "ph" "ps" "ch") "ta") ("syl3anc" ("w3a" "ph" "ps" "ch") "ph" "ps" "th" "ta" ("simp1" "ph" "ps" "ch") ("simp2" "ph" "ps" "ch") "syld3an3_1" "syld3an3_2")) ;; A syllogism inference. (Contributed by NM, 7-Jul-2008.) (theorem "syld3an1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syld3an1_1" ("wi" ("w3a" "ch" "ps" "th") "ph")) ("syld3an1_2" ("wi" ("w3a" "ph" "ps" "th") "ta"))) (for) ("wi" ("w3a" "ch" "ps" "th") "ta") ("_3com13" "th" "ps" "ch" "ta" ("syld3an3" "th" "ps" "ch" "ph" "ta" ("_3com13" "ch" "ps" "th" "ph" "syld3an1_1") ("_3com13" "ph" "ps" "th" "ta" "syld3an1_2")))) ;; A syllogism inference. (Contributed by NM, 20-May-2007.) (theorem "syld3an2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("syld3an2_1" ("wi" ("w3a" "ph" "ch" "th") "ps")) ("syld3an2_2" ("wi" ("w3a" "ph" "ps" "th") "ta"))) (for) ("wi" ("w3a" "ph" "ch" "th") "ta") ("_3com23" "ph" "th" "ch" "ta" ("syld3an3" "ph" "th" "ch" "ps" "ta" ("_3com23" "ph" "ch" "th" "ps" "syld3an2_1") ("_3com23" "ph" "ps" "th" "ta" "syld3an2_2")))) ;; A syllogism inference. (Contributed by NM, 24-Feb-2005.) (theorem "syl3anl1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl3anl1_1" ("wi" "ph" "ps")) ("syl3anl1_2" ("wi" ("wa" ("w3a" "ps" "ch" "th") "ta") "et"))) (for) ("wi" ("wa" ("w3a" "ph" "ch" "th") "ta") "et") ("sylan" ("w3a" "ph" "ch" "th") ("w3a" "ps" "ch" "th") "ta" "et" ("_3anim1i" "ph" "ps" "ch" "th" "syl3anl1_1") "syl3anl1_2")) ;; A syllogism inference. (Contributed by NM, 24-Feb-2005.) (theorem "syl3anl2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl3anl2_1" ("wi" "ph" "ch")) ("syl3anl2_2" ("wi" ("wa" ("w3a" "ps" "ch" "th") "ta") "et"))) (for) ("wi" ("wa" ("w3a" "ps" "ph" "th") "ta") "et") ("imp" ("w3a" "ps" "ph" "th") "ta" "et" ("syl3an2" "ph" "ps" "ch" "th" ("wi" "ta" "et") "syl3anl2_1" ("ex" ("w3a" "ps" "ch" "th") "ta" "et" "syl3anl2_2")))) ;; A syllogism inference. (Contributed by NM, 24-Feb-2005.) (theorem "syl3anl3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl3anl3_1" ("wi" "ph" "th")) ("syl3anl3_2" ("wi" ("wa" ("w3a" "ps" "ch" "th") "ta") "et"))) (for) ("wi" ("wa" ("w3a" "ps" "ch" "ph") "ta") "et") ("sylan" ("w3a" "ps" "ch" "ph") ("w3a" "ps" "ch" "th") "ta" "et" ("_3anim3i" "ph" "th" "ps" "ch" "syl3anl3_1") "syl3anl3_2")) ;; A triple syllogism inference. (Contributed by NM, 24-Dec-2006.) (theorem "syl3anl" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff"))) (for ("syl3anl_1" ("wi" "ph" "ps")) ("syl3anl_2" ("wi" "ch" "th")) ("syl3anl_3" ("wi" "ta" "et")) ("syl3anl_4" ("wi" ("wa" ("w3a" "ps" "th" "et") "ze") "si"))) (for) ("wi" ("wa" ("w3a" "ph" "ch" "ta") "ze") "si") ("sylan" ("w3a" "ph" "ch" "ta") ("w3a" "ps" "th" "et") "ze" "si" ("_3anim123i" "ph" "ps" "ch" "th" "ta" "et" "syl3anl_1" "syl3anl_2" "syl3anl_3") "syl3anl_4")) ;; A syllogism inference. (Contributed by NM, 31-Jul-2007.) (theorem "syl3anr1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl3anr1_1" ("wi" "ph" "ps")) ("syl3anr1_2" ("wi" ("wa" "ch" ("w3a" "ps" "th" "ta")) "et"))) (for) ("wi" ("wa" "ch" ("w3a" "ph" "th" "ta")) "et") ("sylan2" ("w3a" "ph" "th" "ta") "ch" ("w3a" "ps" "th" "ta") "et" ("_3anim1i" "ph" "ps" "th" "ta" "syl3anr1_1") "syl3anr1_2")) ;; A syllogism inference. (Contributed by NM, 1-Aug-2007.) (theorem "syl3anr2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl3anr2_1" ("wi" "ph" "th")) ("syl3anr2_2" ("wi" ("wa" "ch" ("w3a" "ps" "th" "ta")) "et"))) (for) ("wi" ("wa" "ch" ("w3a" "ps" "ph" "ta")) "et") ("ancoms" ("w3a" "ps" "ph" "ta") "ch" "et" ("syl3anl2" "ph" "ps" "th" "ta" "ch" "et" "syl3anr2_1" ("ancoms" "ch" ("w3a" "ps" "th" "ta") "et" "syl3anr2_2")))) ;; A syllogism inference. (Contributed by NM, 23-Aug-2007.) (theorem "syl3anr3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl3anr3_1" ("wi" "ph" "ta")) ("syl3anr3_2" ("wi" ("wa" "ch" ("w3a" "ps" "th" "ta")) "et"))) (for) ("wi" ("wa" "ch" ("w3a" "ps" "th" "ph")) "et") ("sylan2" ("w3a" "ps" "th" "ph") "ch" ("w3a" "ps" "th" "ta") "et" ("_3anim3i" "ph" "ta" "ps" "th" "syl3anr3_1") "syl3anr3_2")) ;; Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.) (theorem "_3impdi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3impdi_1" ("wi" ("wa" ("wa" "ph" "ps") ("wa" "ph" "ch")) "th"))) (for) ("wi" ("w3a" "ph" "ps" "ch") "th") ("_3impb" "ph" "ps" "ch" "th" ("anandis" "ph" "ps" "ch" "th" "_3impdi_1"))) ;; Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.) (theorem "_3impdir" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3impdir_1" ("wi" ("wa" ("wa" "ph" "ps") ("wa" "ch" "ps")) "th"))) (for) ("wi" ("w3a" "ph" "ch" "ps") "th") ("_3impa" "ph" "ch" "ps" "th" ("anandirs" "ph" "ch" "ps" "th" "_3impdir_1"))) ;; Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) (theorem "_3anidm12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_3anidm12_1" ("wi" ("w3a" "ph" "ph" "ps") "ch"))) (for) ("wi" ("wa" "ph" "ps") "ch") ("anabsi5" "ph" "ps" "ch" ("_3expib" "ph" "ph" "ps" "ch" "_3anidm12_1"))) ;; Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) (theorem "_3anidm13" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_3anidm13_1" ("wi" ("w3a" "ph" "ps" "ph") "ch"))) (for) ("wi" ("wa" "ph" "ps") "ch") ("_3anidm12" "ph" "ps" "ch" ("_3com23" "ph" "ps" "ph" "ch" "_3anidm13_1"))) ;; Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.) (theorem "_3anidm23" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_3anidm23_1" ("wi" ("w3a" "ph" "ps" "ps") "ch"))) (for) ("wi" ("wa" "ph" "ps") "ch") ("anabss3" "ph" "ps" "ch" ("_3expa" "ph" "ps" "ps" "ch" "_3anidm23_1"))) ;; ~ syl3an with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) (theorem "syl2an3an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl2an3an_1" ("wi" "ph" "ps")) ("syl2an3an_2" ("wi" "ph" "ch")) ("syl2an3an_3" ("wi" "th" "ta")) ("syl2an3an_4" ("wi" ("w3a" "ps" "ch" "ta") "et"))) (for) ("wi" ("wa" "ph" "th") "et") ("_3anidm12" "ph" "th" "et" ("syl3an" "ph" "ps" "ph" "ch" "th" "ta" "et" "syl2an3an_1" "syl2an3an_2" "syl2an3an_3" "syl2an3an_4"))) ;; Deduction related to ~ syl3an with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) (theorem "syl2an23an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("syl2an23an_1" ("wi" "ph" "ps")) ("syl2an23an_2" ("wi" "ph" "ch")) ("syl2an23an_3" ("wi" ("wa" "th" "ph") "ta")) ("syl2an23an_4" ("wi" ("w3a" "ps" "ch" "ta") "et"))) (for) ("wi" ("wa" "th" "ph") "et") ("anabsi7" "th" "ph" "et" ("syl5" ("wa" "th" "ph") "ta" "ph" "et" "syl2an23an_3" ("sylc" "ph" "ps" "ch" ("wi" "ta" "et") "syl2an23an_1" "syl2an23an_2" ("_3exp" "ps" "ch" "ta" "et" "syl2an23an_4"))))) ;; Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.) (theorem "_3ori" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("_3ori_1" ("w3o" "ph" "ps" "ch"))) (for) ("wi" ("wa" ("wn" "ph") ("wn" "ps")) "ch") ("sylbir" ("wa" ("wn" "ph") ("wn" "ps")) ("wn" ("wo" "ph" "ps")) "ch" ("ioran" "ph" "ps") ("ori" ("wo" "ph" "ps") "ch" ("mpbi" ("w3o" "ph" "ps" "ch") ("wo" ("wo" "ph" "ps") "ch") "_3ori_1" ("df_3or" "ph" "ps" "ch"))))) ;; Disjunction of three antecedents. (Contributed by NM, 8-Apr-1994.) (theorem "_3jao" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("w3a" ("wi" "ph" "ps") ("wi" "ch" "ps") ("wi" "th" "ps")) ("wi" ("w3o" "ph" "ch" "th") "ps")) ("syl5bi" ("w3o" "ph" "ch" "th") ("wo" ("wo" "ph" "ch") "th") ("w3a" ("wi" "ph" "ps") ("wi" "ch" "ps") ("wi" "th" "ps")) "ps" ("df_3or" "ph" "ch" "th") ("_3imp" ("wi" "ph" "ps") ("wi" "ch" "ps") ("wi" "th" "ps") ("wi" ("wo" ("wo" "ph" "ch") "th") "ps") ("syl6" ("wi" "ph" "ps") ("wi" "ch" "ps") ("wi" ("wo" "ph" "ch") "ps") ("wi" ("wi" "th" "ps") ("wi" ("wo" ("wo" "ph" "ch") "th") "ps")) ("jao" "ph" "ps" "ch") ("jao" ("wo" "ph" "ch") "ps" "th"))))) ;; Disjunction of three antecedents. (Contributed by NM, 13-Sep-2011.) (theorem "_3jaob" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wi" ("w3o" "ph" "ch" "th") "ps") ("w3a" ("wi" "ph" "ps") ("wi" "ch" "ps") ("wi" "th" "ps"))) ("impbii" ("wi" ("w3o" "ph" "ch" "th") "ps") ("w3a" ("wi" "ph" "ps") ("wi" "ch" "ps") ("wi" "th" "ps")) ("_3jca" ("wi" ("w3o" "ph" "ch" "th") "ps") ("wi" "ph" "ps") ("wi" "ch" "ps") ("wi" "th" "ps") ("imim1i" "ph" ("w3o" "ph" "ch" "th") "ps" ("_3mix1" "ph" "ch" "th")) ("imim1i" "ch" ("w3o" "ph" "ch" "th") "ps" ("_3mix2" "ch" "ph" "th")) ("imim1i" "th" ("w3o" "ph" "ch" "th") "ps" ("_3mix3" "th" "ph" "ch"))) ("_3jao" "ph" "ps" "ch" "th"))) ;; Disjunction of three antecedents (inference). (Contributed by NM, 12-Sep-1995.) (theorem "_3jaoi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3jaoi_1" ("wi" "ph" "ps")) ("_3jaoi_2" ("wi" "ch" "ps")) ("_3jaoi_3" ("wi" "th" "ps"))) (for) ("wi" ("w3o" "ph" "ch" "th") "ps") ("ax_mp" ("w3a" ("wi" "ph" "ps") ("wi" "ch" "ps") ("wi" "th" "ps")) ("wi" ("w3o" "ph" "ch" "th") "ps") ("_3pm3_2i" ("wi" "ph" "ps") ("wi" "ch" "ps") ("wi" "th" "ps") "_3jaoi_1" "_3jaoi_2" "_3jaoi_3") ("_3jao" "ph" "ps" "ch" "th"))) ;; Disjunction of three antecedents (deduction). (Contributed by NM, 14-Oct-2005.) (theorem "_3jaod" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3jaod_1" ("wi" "ph" ("wi" "ps" "ch"))) ("_3jaod_2" ("wi" "ph" ("wi" "th" "ch"))) ("_3jaod_3" ("wi" "ph" ("wi" "ta" "ch")))) (for) ("wi" "ph" ("wi" ("w3o" "ps" "th" "ta") "ch")) ("syl3anc" "ph" ("wi" "ps" "ch") ("wi" "th" "ch") ("wi" "ta" "ch") ("wi" ("w3o" "ps" "th" "ta") "ch") "_3jaod_1" "_3jaod_2" "_3jaod_3" ("_3jao" "ps" "ch" "th" "ta"))) ;; Disjunction of three antecedents (inference). (Contributed by NM, 14-Oct-2005.) (theorem "_3jaoian" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3jaoian_1" ("wi" ("wa" "ph" "ps") "ch")) ("_3jaoian_2" ("wi" ("wa" "th" "ps") "ch")) ("_3jaoian_3" ("wi" ("wa" "ta" "ps") "ch"))) (for) ("wi" ("wa" ("w3o" "ph" "th" "ta") "ps") "ch") ("imp" ("w3o" "ph" "th" "ta") "ps" "ch" ("_3jaoi" "ph" ("wi" "ps" "ch") "th" "ta" ("ex" "ph" "ps" "ch" "_3jaoian_1") ("ex" "th" "ps" "ch" "_3jaoian_2") ("ex" "ta" "ps" "ch" "_3jaoian_3")))) ;; Disjunction of three antecedents (deduction). (Contributed by NM, 14-Oct-2005.) (theorem "_3jaodan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3jaodan_1" ("wi" ("wa" "ph" "ps") "ch")) ("_3jaodan_2" ("wi" ("wa" "ph" "th") "ch")) ("_3jaodan_3" ("wi" ("wa" "ph" "ta") "ch"))) (for) ("wi" ("wa" "ph" ("w3o" "ps" "th" "ta")) "ch") ("imp" "ph" ("w3o" "ps" "th" "ta") "ch" ("_3jaod" "ph" "ps" "ch" "th" "ta" ("ex" "ph" "ps" "ch" "_3jaodan_1") ("ex" "ph" "th" "ch" "_3jaodan_2") ("ex" "ph" "ta" "ch" "_3jaodan_3")))) ;; Eliminate a three-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.) (theorem "mpjao3dan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mpjao3dan_1" ("wi" ("wa" "ph" "ps") "ch")) ("mpjao3dan_2" ("wi" ("wa" "ph" "th") "ch")) ("mpjao3dan_3" ("wi" ("wa" "ph" "ta") "ch")) ("mpjao3dan_4" ("wi" "ph" ("w3o" "ps" "th" "ta")))) (for) ("wi" "ph" "ch") ("mpjaodan" "ph" ("wo" "ps" "th") "ch" "ta" ("jaodan" "ph" "ps" "ch" "th" "mpjao3dan_1" "mpjao3dan_2") "mpjao3dan_3" ("sylib" "ph" ("w3o" "ps" "th" "ta") ("wo" ("wo" "ps" "th") "ta") "mpjao3dan_4" ("df_3or" "ps" "th" "ta")))) ;; Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "_3jaao" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("_3jaao_1" ("wi" "ph" ("wi" "ps" "ch"))) ("_3jaao_2" ("wi" "th" ("wi" "ta" "ch"))) ("_3jaao_3" ("wi" "et" ("wi" "ze" "ch")))) (for) ("wi" ("w3a" "ph" "th" "et") ("wi" ("w3o" "ps" "ta" "ze") "ch")) ("_3jaod" ("w3a" "ph" "th" "et") "ps" "ch" "ta" "ze" ("_3ad2ant1" "ph" "th" ("wi" "ps" "ch") "et" "_3jaao_1") ("_3ad2ant2" "th" "ph" ("wi" "ta" "ch") "et" "_3jaao_2") ("_3ad2ant3" "et" "ph" ("wi" "ze" "ch") "th" "_3jaao_3"))) ;; Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.) (theorem "syl3an9b" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("syl3an9b_1" ("wi" "ph" ("wb" "ps" "ch"))) ("syl3an9b_2" ("wi" "th" ("wb" "ch" "ta"))) ("syl3an9b_3" ("wi" "et" ("wb" "ta" "ze")))) (for) ("wi" ("w3a" "ph" "th" "et") ("wb" "ps" "ze")) ("_3impa" "ph" "th" "et" ("wb" "ps" "ze") ("sylan9bb" ("wa" "ph" "th") "ps" "ta" "et" "ze" ("sylan9bb" "ph" "ps" "ch" "th" "ta" "syl3an9b_1" "syl3an9b_2") "syl3an9b_3"))) ;; Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.) (theorem "_3orbi123d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("bi3d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("bi3d_2" ("wi" "ph" ("wb" "th" "ta"))) ("bi3d_3" ("wi" "ph" ("wb" "et" "ze")))) (for) ("wi" "ph" ("wb" ("w3o" "ps" "th" "et") ("w3o" "ch" "ta" "ze"))) ("_3bitr4g" "ph" ("wo" ("wo" "ps" "th") "et") ("wo" ("wo" "ch" "ta") "ze") ("w3o" "ps" "th" "et") ("w3o" "ch" "ta" "ze") ("orbi12d" "ph" ("wo" "ps" "th") ("wo" "ch" "ta") "et" "ze" ("orbi12d" "ph" "ps" "ch" "th" "ta" "bi3d_1" "bi3d_2") "bi3d_3") ("df_3or" "ps" "th" "et") ("df_3or" "ch" "ta" "ze"))) ;; Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.) (theorem "_3anbi123d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("bi3d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("bi3d_2" ("wi" "ph" ("wb" "th" "ta"))) ("bi3d_3" ("wi" "ph" ("wb" "et" "ze")))) (for) ("wi" "ph" ("wb" ("w3a" "ps" "th" "et") ("w3a" "ch" "ta" "ze"))) ("_3bitr4g" "ph" ("wa" ("wa" "ps" "th") "et") ("wa" ("wa" "ch" "ta") "ze") ("w3a" "ps" "th" "et") ("w3a" "ch" "ta" "ze") ("anbi12d" "ph" ("wa" "ps" "th") ("wa" "ch" "ta") "et" "ze" ("anbi12d" "ph" "ps" "ch" "th" "ta" "bi3d_1" "bi3d_2") "bi3d_3") ("df_3an" "ps" "th" "et") ("df_3an" "ch" "ta" "ze"))) ;; Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) (theorem "_3anbi12d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("_3anbi12d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3anbi12d_2" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" ("w3a" "ps" "th" "et") ("w3a" "ch" "ta" "et"))) ("_3anbi123d" "ph" "ps" "ch" "th" "ta" "et" "et" "_3anbi12d_1" "_3anbi12d_2" ("biidd" "ph" "et"))) ;; Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) (theorem "_3anbi13d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("_3anbi12d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3anbi12d_2" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" ("w3a" "ps" "et" "th") ("w3a" "ch" "et" "ta"))) ("_3anbi123d" "ph" "ps" "ch" "et" "et" "th" "ta" "_3anbi12d_1" ("biidd" "ph" "et") "_3anbi12d_2")) ;; Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) (theorem "_3anbi23d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("_3anbi12d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("_3anbi12d_2" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" ("w3a" "et" "ps" "th") ("w3a" "et" "ch" "ta"))) ("_3anbi123d" "ph" "et" "et" "ps" "ch" "th" "ta" ("biidd" "ph" "et") "_3anbi12d_1" "_3anbi12d_2")) ;; Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) (theorem "_3anbi1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3anbi1d_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("w3a" "ps" "th" "ta") ("w3a" "ch" "th" "ta"))) ("_3anbi12d" "ph" "ps" "ch" "th" "th" "ta" "_3anbi1d_1" ("biidd" "ph" "th"))) ;; Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) (theorem "_3anbi2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3anbi1d_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("w3a" "th" "ps" "ta") ("w3a" "th" "ch" "ta"))) ("_3anbi12d" "ph" "th" "th" "ps" "ch" "ta" ("biidd" "ph" "th") "_3anbi1d_1")) ;; Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) (theorem "_3anbi3d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3anbi1d_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("w3a" "th" "ta" "ps") ("w3a" "th" "ta" "ch"))) ("_3anbi13d" "ph" "th" "th" "ps" "ch" "ta" ("biidd" "ph" "th") "_3anbi1d_1")) ;; Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.) (theorem "_3anim123d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("_3anim123d_1" ("wi" "ph" ("wi" "ps" "ch"))) ("_3anim123d_2" ("wi" "ph" ("wi" "th" "ta"))) ("_3anim123d_3" ("wi" "ph" ("wi" "et" "ze")))) (for) ("wi" "ph" ("wi" ("w3a" "ps" "th" "et") ("w3a" "ch" "ta" "ze"))) ("_3imtr4g" "ph" ("wa" ("wa" "ps" "th") "et") ("wa" ("wa" "ch" "ta") "ze") ("w3a" "ps" "th" "et") ("w3a" "ch" "ta" "ze") ("anim12d" "ph" ("wa" "ps" "th") ("wa" "ch" "ta") "et" "ze" ("anim12d" "ph" "ps" "ch" "th" "ta" "_3anim123d_1" "_3anim123d_2") "_3anim123d_3") ("df_3an" "ps" "th" "et") ("df_3an" "ch" "ta" "ze"))) ;; Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.) (theorem "_3orim123d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("_3anim123d_1" ("wi" "ph" ("wi" "ps" "ch"))) ("_3anim123d_2" ("wi" "ph" ("wi" "th" "ta"))) ("_3anim123d_3" ("wi" "ph" ("wi" "et" "ze")))) (for) ("wi" "ph" ("wi" ("w3o" "ps" "th" "et") ("w3o" "ch" "ta" "ze"))) ("_3imtr4g" "ph" ("wo" ("wo" "ps" "th") "et") ("wo" ("wo" "ch" "ta") "ze") ("w3o" "ps" "th" "et") ("w3o" "ch" "ta" "ze") ("orim12d" "ph" ("wo" "ps" "th") ("wo" "ch" "ta") "et" "ze" ("orim12d" "ph" "ps" "ch" "th" "ta" "_3anim123d_1" "_3anim123d_2") "_3anim123d_3") ("df_3or" "ps" "th" "et") ("df_3or" "ch" "ta" "ze"))) ;; Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.) (theorem "an6" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wb" ("wa" ("w3a" "ph" "ps" "ch") ("w3a" "th" "ta" "et")) ("w3a" ("wa" "ph" "th") ("wa" "ps" "ta") ("wa" "ch" "et"))) ("_3bitr4i" ("wa" ("wa" ("wa" "ph" "ps") "ch") ("wa" ("wa" "th" "ta") "et")) ("wa" ("wa" ("wa" "ph" "th") ("wa" "ps" "ta")) ("wa" "ch" "et")) ("wa" ("w3a" "ph" "ps" "ch") ("w3a" "th" "ta" "et")) ("w3a" ("wa" "ph" "th") ("wa" "ps" "ta") ("wa" "ch" "et")) ("bitri" ("wa" ("wa" ("wa" "ph" "ps") "ch") ("wa" ("wa" "th" "ta") "et")) ("wa" ("wa" ("wa" "ph" "ps") ("wa" "th" "ta")) ("wa" "ch" "et")) ("wa" ("wa" ("wa" "ph" "th") ("wa" "ps" "ta")) ("wa" "ch" "et")) ("an4" ("wa" "ph" "ps") "ch" ("wa" "th" "ta") "et") ("anbi1i" ("wa" ("wa" "ph" "ps") ("wa" "th" "ta")) ("wa" ("wa" "ph" "th") ("wa" "ps" "ta")) ("wa" "ch" "et") ("an4" "ph" "ps" "th" "ta"))) ("anbi12i" ("w3a" "ph" "ps" "ch") ("wa" ("wa" "ph" "ps") "ch") ("w3a" "th" "ta" "et") ("wa" ("wa" "th" "ta") "et") ("df_3an" "ph" "ps" "ch") ("df_3an" "th" "ta" "et")) ("df_3an" ("wa" "ph" "th") ("wa" "ps" "ta") ("wa" "ch" "et")))) ;; Analogue of ~ an4 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) (theorem "_3an6" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wb" ("w3a" ("wa" "ph" "ps") ("wa" "ch" "th") ("wa" "ta" "et")) ("wa" ("w3a" "ph" "ch" "ta") ("w3a" "ps" "th" "et"))) ("bicomi" ("wa" ("w3a" "ph" "ch" "ta") ("w3a" "ps" "th" "et")) ("w3a" ("wa" "ph" "ps") ("wa" "ch" "th") ("wa" "ta" "et")) ("an6" "ph" "ch" "ta" "ps" "th" "et"))) ;; Analogue of ~ or4 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (theorem "_3or6" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for) (for) ("wb" ("w3o" ("wo" "ph" "ps") ("wo" "ch" "th") ("wo" "ta" "et")) ("wo" ("w3o" "ph" "ch" "ta") ("w3o" "ps" "th" "et"))) ("_3bitr4i" ("wo" ("wo" ("wo" "ph" "ps") ("wo" "ch" "th")) ("wo" "ta" "et")) ("wo" ("wo" ("wo" "ph" "ch") "ta") ("wo" ("wo" "ps" "th") "et")) ("w3o" ("wo" "ph" "ps") ("wo" "ch" "th") ("wo" "ta" "et")) ("wo" ("w3o" "ph" "ch" "ta") ("w3o" "ps" "th" "et")) ("bitr2i" ("wo" ("wo" ("wo" "ph" "ch") "ta") ("wo" ("wo" "ps" "th") "et")) ("wo" ("wo" ("wo" "ph" "ch") ("wo" "ps" "th")) ("wo" "ta" "et")) ("wo" ("wo" ("wo" "ph" "ps") ("wo" "ch" "th")) ("wo" "ta" "et")) ("or4" ("wo" "ph" "ch") "ta" ("wo" "ps" "th") "et") ("orbi1i" ("wo" ("wo" "ph" "ch") ("wo" "ps" "th")) ("wo" ("wo" "ph" "ps") ("wo" "ch" "th")) ("wo" "ta" "et") ("or4" "ph" "ch" "ps" "th"))) ("df_3or" ("wo" "ph" "ps") ("wo" "ch" "th") ("wo" "ta" "et")) ("orbi12i" ("w3o" "ph" "ch" "ta") ("wo" ("wo" "ph" "ch") "ta") ("w3o" "ps" "th" "et") ("wo" ("wo" "ps" "th") "et") ("df_3or" "ph" "ch" "ta") ("df_3or" "ps" "th" "et")))) ;; An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) (theorem "mp3an1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mp3an1_1" "ph") ("mp3an1_2" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" "ps" "ch") "th") ("mpan" "ph" ("wa" "ps" "ch") "th" "mp3an1_1" ("_3expb" "ph" "ps" "ch" "th" "mp3an1_2"))) ;; An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) (theorem "mp3an2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mp3an2_1" "ps") ("mp3an2_2" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" "ch") "th") ("mpanl2" "ph" "ps" "ch" "th" "mp3an2_1" ("_3expa" "ph" "ps" "ch" "th" "mp3an2_2"))) ;; An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) (theorem "mp3an3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mp3an3_1" "ch") ("mp3an3_2" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" "ps") "th") ("mpi" ("wa" "ph" "ps") "ch" "th" "mp3an3_1" ("_3expia" "ph" "ps" "ch" "th" "mp3an3_2"))) ;; An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.) (theorem "mp3an12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mp3an12_1" "ph") ("mp3an12_2" "ps") ("mp3an12_3" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" "ch" "th") ("mpan" "ps" "ch" "th" "mp3an12_2" ("mp3an1" "ph" "ps" "ch" "th" "mp3an12_1" "mp3an12_3"))) ;; An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) (theorem "mp3an13" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mp3an13_1" "ph") ("mp3an13_2" "ch") ("mp3an13_3" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" "ps" "th") ("mpan" "ph" "ps" "th" "mp3an13_1" ("mp3an3" "ph" "ps" "ch" "th" "mp3an13_2" "mp3an13_3"))) ;; An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) (theorem "mp3an23" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mp3an23_1" "ps") ("mp3an23_2" "ch") ("mp3an23_3" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" "ph" "th") ("mpan2" "ph" "ps" "th" "mp3an23_1" ("mp3an3" "ph" "ps" "ch" "th" "mp3an23_2" "mp3an23_3"))) ;; An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) (theorem "mp3an1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mp3an1i_1" "ps") ("mp3an1i_2" ("wi" "ph" ("wi" ("w3a" "ps" "ch" "th") "ta")))) (for) ("wi" "ph" ("wi" ("wa" "ch" "th") "ta")) ("com12" ("wa" "ch" "th") "ph" "ta" ("mp3an1" "ps" "ch" "th" ("wi" "ph" "ta") "mp3an1i_1" ("com12" "ph" ("w3a" "ps" "ch" "th") "ta" "mp3an1i_2")))) ;; An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) (theorem "mp3anl1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mp3anl1_1" "ph") ("mp3anl1_2" ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta"))) (for) ("wi" ("wa" ("wa" "ps" "ch") "th") "ta") ("imp" ("wa" "ps" "ch") "th" "ta" ("mp3an1" "ph" "ps" "ch" ("wi" "th" "ta") "mp3anl1_1" ("ex" ("w3a" "ph" "ps" "ch") "th" "ta" "mp3anl1_2")))) ;; An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) (theorem "mp3anl2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mp3anl2_1" "ps") ("mp3anl2_2" ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta"))) (for) ("wi" ("wa" ("wa" "ph" "ch") "th") "ta") ("imp" ("wa" "ph" "ch") "th" "ta" ("mp3an2" "ph" "ps" "ch" ("wi" "th" "ta") "mp3anl2_1" ("ex" ("w3a" "ph" "ps" "ch") "th" "ta" "mp3anl2_2")))) ;; An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) (theorem "mp3anl3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mp3anl3_1" "ch") ("mp3anl3_2" ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta"))) (for) ("wi" ("wa" ("wa" "ph" "ps") "th") "ta") ("imp" ("wa" "ph" "ps") "th" "ta" ("mp3an3" "ph" "ps" "ch" ("wi" "th" "ta") "mp3anl3_1" ("ex" ("w3a" "ph" "ps" "ch") "th" "ta" "mp3anl3_2")))) ;; An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) (theorem "mp3anr1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mp3anr1_1" "ps") ("mp3anr1_2" ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ta"))) (for) ("wi" ("wa" "ph" ("wa" "ch" "th")) "ta") ("ancoms" ("wa" "ch" "th") "ph" "ta" ("mp3anl1" "ps" "ch" "th" "ph" "ta" "mp3anr1_1" ("ancoms" "ph" ("w3a" "ps" "ch" "th") "ta" "mp3anr1_2")))) ;; An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.) (theorem "mp3anr2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mp3anr2_1" "ch") ("mp3anr2_2" ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ta"))) (for) ("wi" ("wa" "ph" ("wa" "ps" "th")) "ta") ("ancoms" ("wa" "ps" "th") "ph" "ta" ("mp3anl2" "ps" "ch" "th" "ph" "ta" "mp3anr2_1" ("ancoms" "ph" ("w3a" "ps" "ch" "th") "ta" "mp3anr2_2")))) ;; An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.) (theorem "mp3anr3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mp3anr3_1" "th") ("mp3anr3_2" ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ta"))) (for) ("wi" ("wa" "ph" ("wa" "ps" "ch")) "ta") ("ancoms" ("wa" "ps" "ch") "ph" "ta" ("mp3anl3" "ps" "ch" "th" "ph" "ta" "mp3anr3_1" ("ancoms" "ph" ("w3a" "ps" "ch" "th") "ta" "mp3anr3_2")))) ;; An inference based on modus ponens. (Contributed by NM, 14-May-1999.) (theorem "mp3an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mp3an_1" "ph") ("mp3an_2" "ps") ("mp3an_3" "ch") ("mp3an_4" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) "th" ("mp2an" "ps" "ch" "th" "mp3an_2" "mp3an_3" ("mp3an1" "ph" "ps" "ch" "th" "mp3an_1" "mp3an_4"))) ;; An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) (theorem "mpd3an3" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpd3an3_2" ("wi" ("wa" "ph" "ps") "ch")) ("mpd3an3_3" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" ("wa" "ph" "ps") "th") ("mpdan" ("wa" "ph" "ps") "ch" "th" "mpd3an3_2" ("_3expa" "ph" "ps" "ch" "th" "mpd3an3_3"))) ;; An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) (theorem "mpd3an23" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("mpd3an23_1" ("wi" "ph" "ps")) ("mpd3an23_2" ("wi" "ph" "ch")) ("mpd3an23_3" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) ("wi" "ph" "th") ("syl3anc" "ph" "ph" "ps" "ch" "th" ("id" "ph") "mpd3an23_1" "mpd3an23_2" "mpd3an23_3")) ;; A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.) (theorem "mp3and" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mp3and_1" ("wi" "ph" "ps")) ("mp3and_2" ("wi" "ph" "ch")) ("mp3and_3" ("wi" "ph" "th")) ("mp3and_4" ("wi" "ph" ("wi" ("w3a" "ps" "ch" "th") "ta")))) (for) ("wi" "ph" "ta") ("mpd" "ph" ("w3a" "ps" "ch" "th") "ta" ("_3jca" "ph" "ps" "ch" "th" "mp3and_1" "mp3and_2" "mp3and_3") "mp3and_4")) ;; ~ mp3an with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.) (theorem "mp3an12i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mp3an12i_1" "ph") ("mp3an12i_2" "ps") ("mp3an12i_3" ("wi" "ch" "th")) ("mp3an12i_4" ("wi" ("w3a" "ph" "ps" "th") "ta"))) (for) ("wi" "ch" "ta") ("syl" "ch" "th" "ta" "mp3an12i_3" ("mp3an12" "ph" "ps" "th" "ta" "mp3an12i_1" "mp3an12i_2" "mp3an12i_4"))) ;; ~ mp3an with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) (theorem "mp3an2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("mp3an2i_1" "ph") ("mp3an2i_2" ("wi" "ps" "ch")) ("mp3an2i_3" ("wi" "ps" "th")) ("mp3an2i_4" ("wi" ("w3a" "ph" "ch" "th") "ta"))) (for) ("wi" "ps" "ta") ("syl2anc" "ps" "ch" "th" "ta" "mp3an2i_2" "mp3an2i_3" ("mp3an1" "ph" "ch" "th" "ta" "mp3an2i_1" "mp3an2i_4"))) ;; ~ mp3an with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) (theorem "mp3an3an" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("mp3an3an_1" "ph") ("mp3an3an_2" ("wi" "ps" "ch")) ("mp3an3an_3" ("wi" "th" "ta")) ("mp3an3an_4" ("wi" ("w3a" "ph" "ch" "ta") "et"))) (for) ("wi" ("wa" "ps" "th") "et") ("syl2an" "ps" "ch" "ta" "et" "th" "mp3an3an_2" "mp3an3an_3" ("mp3an1" "ph" "ch" "ta" "et" "mp3an3an_1" "mp3an3an_4"))) ;; An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) (theorem "mp3an2ani" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("mp3an2ani_1" "ph") ("mp3an2ani_2" ("wi" "ps" "ch")) ("mp3an2ani_3" ("wi" ("wa" "ps" "th") "ta")) ("mp3an2ani_4" ("wi" ("w3a" "ph" "ch" "ta") "et"))) (for) ("wi" ("wa" "ps" "th") "et") ("anabss5" "ps" "th" "et" ("mp3an3an" "ph" "ps" "ch" ("wa" "ps" "th") "ta" "et" "mp3an2ani_1" "mp3an2ani_2" "mp3an2ani_3" "mp3an2ani_4"))) ;; Infer implication from a logical equivalence. Similar to ~ biimpa . (Contributed by NM, 4-Sep-2005.) (theorem "biimp3a" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("biimp3a_1" ("wi" ("wa" "ph" "ps") ("wb" "ch" "th")))) (for) ("wi" ("w3a" "ph" "ps" "ch") "th") ("_3impa" "ph" "ps" "ch" "th" ("biimpa" ("wa" "ph" "ps") "ch" "th" "biimp3a_1"))) ;; Infer implication from a logical equivalence. Similar to ~ biimpar . (Contributed by NM, 2-Jan-2009.) (theorem "biimp3ar" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("biimp3a_1" ("wi" ("wa" "ph" "ps") ("wb" "ch" "th")))) (for) ("wi" ("w3a" "ph" "ps" "th") "ch") ("_3imp" "ph" "ps" "th" "ch" ("exbiri" "ph" "ps" "ch" "th" "biimp3a_1"))) ;; Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.) (theorem "_3anandis" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3anandis_1" ("wi" ("w3a" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ph" "th")) "ta"))) (for) ("wi" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ta") ("syl222anc" ("wa" "ph" ("w3a" "ps" "ch" "th")) "ph" "ps" "ph" "ch" "ph" "th" "ta" ("simpl" "ph" ("w3a" "ps" "ch" "th")) ("simpr1" "ph" "ps" "ch" "th") ("simpl" "ph" ("w3a" "ps" "ch" "th")) ("simpr2" "ph" "ps" "ch" "th") ("simpl" "ph" ("w3a" "ps" "ch" "th")) ("simpr3" "ph" "ps" "ch" "th") "_3anandis_1")) ;; Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (theorem "_3anandirs" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3anandirs_1" ("wi" ("w3a" ("wa" "ph" "th") ("wa" "ps" "th") ("wa" "ch" "th")) "ta"))) (for) ("wi" ("wa" ("w3a" "ph" "ps" "ch") "th") "ta") ("syl222anc" ("wa" ("w3a" "ph" "ps" "ch") "th") "ph" "th" "ps" "th" "ch" "th" "ta" ("simpl1" "ph" "ps" "ch" "th") ("simpr" ("w3a" "ph" "ps" "ch") "th") ("simpl2" "ph" "ps" "ch" "th") ("simpr" ("w3a" "ph" "ps" "ch") "th") ("simpl3" "ph" "ps" "ch" "th") ("simpr" ("w3a" "ph" "ps" "ch") "th") "_3anandirs_1")) ;; Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.) (theorem "ecase23d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("ecase23d_1" ("wi" "ph" ("wn" "ch"))) ("ecase23d_2" ("wi" "ph" ("wn" "th"))) ("ecase23d_3" ("wi" "ph" ("w3o" "ps" "ch" "th")))) (for) ("wi" "ph" "ps") ("mt3d" "ph" "ps" ("wo" "ch" "th") ("sylanbrc" "ph" ("wn" "ch") ("wn" "th") ("wn" ("wo" "ch" "th")) "ecase23d_1" "ecase23d_2" ("ioran" "ch" "th")) ("ord" "ph" "ps" ("wo" "ch" "th") ("sylib" "ph" ("w3o" "ps" "ch" "th") ("wo" "ps" ("wo" "ch" "th")) "ecase23d_3" ("_3orass" "ps" "ch" "th"))))) ;; Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.) (theorem "_3ecase" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3ecase_1" ("wi" ("wn" "ph") "th")) ("_3ecase_2" ("wi" ("wn" "ps") "th")) ("_3ecase_3" ("wi" ("wn" "ch") "th")) ("_3ecase_4" ("wi" ("w3a" "ph" "ps" "ch") "th"))) (for) "th" ("pm2_61nii" "ps" "ch" "th" ("pm2_61i" "ph" ("wi" "ps" ("wi" "ch" "th")) ("_3exp" "ph" "ps" "ch" "th" "_3ecase_4") ("_2a1d" ("wn" "ph") "th" "ps" "ch" "_3ecase_1")) "_3ecase_2" "_3ecase_3")) ;; A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to ~ biorf . (Contributed by Alexander van der Vekens, 8-Sep-2017.) (theorem "_3bior1fd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3biorfd_1" ("wi" "ph" ("wn" "th")))) (for) ("wi" "ph" ("wb" ("wo" "ch" "ps") ("w3o" "th" "ch" "ps"))) ("syl6bbr" "ph" ("wo" "ch" "ps") ("wo" "th" ("wo" "ch" "ps")) ("w3o" "th" "ch" "ps") ("syl" "ph" ("wn" "th") ("wb" ("wo" "ch" "ps") ("wo" "th" ("wo" "ch" "ps"))) "_3biorfd_1" ("biorf" "th" ("wo" "ch" "ps"))) ("_3orass" "th" "ch" "ps"))) ;; A disjunction is equivalent to a threefold disjunction with single falsehood of a conjunction. (Contributed by Alexander van der Vekens, 8-Sep-2017.) (theorem "_3bior1fand" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("_3biorfd_1" ("wi" "ph" ("wn" "th")))) (for) ("wi" "ph" ("wb" ("wo" "ch" "ps") ("w3o" ("wa" "th" "ta") "ch" "ps"))) ("_3bior1fd" "ph" "ps" "ch" ("wa" "th" "ta") ("intnanrd" "ph" "th" "ta" "_3biorfd_1"))) ;; A wff is equivalent to its threefold disjunction with double falsehood, analogous to ~ biorf . (Contributed by Alexander van der Vekens, 8-Sep-2017.) (theorem "_3bior2fd" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3biorfd_1" ("wi" "ph" ("wn" "th"))) ("_3biorfd_2" ("wi" "ph" ("wn" "ch")))) (for) ("wi" "ph" ("wb" "ps" ("w3o" "th" "ch" "ps"))) ("bitrd" "ph" "ps" ("wo" "ch" "ps") ("w3o" "th" "ch" "ps") ("syl" "ph" ("wn" "ch") ("wb" "ps" ("wo" "ch" "ps")) "_3biorfd_2" ("biorf" "ch" "ps")) ("_3bior1fd" "ph" "ps" "ch" "th" "_3biorfd_1"))) ;; A conjunction is equivalent to a threefold conjunction with single truth, analogous to ~ biantrud . (Contributed by Alexander van der Vekens, 26-Sep-2017.) (theorem "_3biant1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("_3biantd_1" ("wi" "ph" "th"))) (for) ("wi" "ph" ("wb" ("wa" "ch" "ps") ("w3a" "th" "ch" "ps"))) ("syl6bbr" "ph" ("wa" "ch" "ps") ("wa" "th" ("wa" "ch" "ps")) ("w3a" "th" "ch" "ps") ("biantrurd" "ph" "th" ("wa" "ch" "ps") "_3biantd_1") ("_3anass" "th" "ch" "ps"))) ;; Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (theorem "intn3an1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("intn3and_1" ("wi" "ph" ("wn" "ps")))) (for) ("wi" "ph" ("wn" ("w3a" "ps" "ch" "th"))) ("nsyl" "ph" "ps" ("w3a" "ps" "ch" "th") "intn3and_1" ("simp1" "ps" "ch" "th"))) ;; Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (theorem "intn3an2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("intn3and_1" ("wi" "ph" ("wn" "ps")))) (for) ("wi" "ph" ("wn" ("w3a" "ch" "ps" "th"))) ("nsyl" "ph" "ps" ("w3a" "ch" "ps" "th") "intn3and_1" ("simp2" "ch" "ps" "th"))) ;; Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (theorem "intn3an3d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("intn3and_1" ("wi" "ph" ("wn" "ps")))) (for) ("wi" "ph" ("wn" ("w3a" "ch" "th" "ps"))) ("nsyl" "ph" "ps" ("w3a" "ch" "th" "ps") "intn3and_1" ("simp3" "ch" "th" "ps"))) ;; Distribution of conjunction over threefold conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.) (theorem "an3andi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("w3a" "ps" "ch" "th")) ("w3a" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ph" "th"))) ("bitr4i" ("wa" "ph" ("w3a" "ps" "ch" "th")) ("wa" ("wa" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("wa" "ph" "th")) ("w3a" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ph" "th")) ("_3bitri" ("wa" "ph" ("w3a" "ps" "ch" "th")) ("wa" "ph" ("wa" ("wa" "ps" "ch") "th")) ("wa" ("wa" "ph" ("wa" "ps" "ch")) ("wa" "ph" "th")) ("wa" ("wa" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("wa" "ph" "th")) ("anbi2i" ("w3a" "ps" "ch" "th") ("wa" ("wa" "ps" "ch") "th") "ph" ("df_3an" "ps" "ch" "th")) ("anandi" "ph" ("wa" "ps" "ch") "th") ("anbi1i" ("wa" "ph" ("wa" "ps" "ch")) ("wa" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("wa" "ph" "th") ("anandi" "ph" "ps" "ch"))) ("df_3an" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ph" "th")))) ;; Rearrange a 9-fold conjunction. (Contributed by Thierry Arnoux, 14-Apr-2019.) (theorem "an33rean" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff")) ("si" ( "wff")) ("rh" ( "wff"))) (for) (for) ("wb" ("w3a" ("w3a" "ph" "ps" "ch") ("w3a" "th" "ta" "et") ("w3a" "ze" "si" "rh")) ("wa" ("w3a" "ph" "ta" "rh") ("w3a" ("wa" "ps" "th") ("wa" "et" "si") ("wa" "ch" "ze")))) ("_3bitri" ("w3a" ("w3a" "ph" "ps" "ch") ("w3a" "th" "ta" "et") ("w3a" "ze" "si" "rh")) ("w3a" ("wa" "ph" ("wa" "ps" "ch")) ("wa" "ta" ("wa" "th" "et")) ("wa" "rh" ("wa" "si" "ze"))) ("wa" ("w3a" "ph" "ta" "rh") ("w3a" ("wa" "ps" "ch") ("wa" "th" "et") ("wa" "si" "ze"))) ("wa" ("w3a" "ph" "ta" "rh") ("w3a" ("wa" "ps" "th") ("wa" "et" "si") ("wa" "ch" "ze"))) ("_3anbi123i" ("w3a" "ph" "ps" "ch") ("wa" "ph" ("wa" "ps" "ch")) ("w3a" "th" "ta" "et") ("wa" "ta" ("wa" "th" "et")) ("w3a" "ze" "si" "rh") ("wa" "rh" ("wa" "si" "ze")) ("_3anass" "ph" "ps" "ch") ("_3anan12" "th" "ta" "et") ("bitri" ("w3a" "ze" "si" "rh") ("w3a" "rh" "si" "ze") ("wa" "rh" ("wa" "si" "ze")) ("_3anrev" "ze" "si" "rh") ("_3anass" "rh" "si" "ze"))) ("_3an6" "ph" ("wa" "ps" "ch") "ta" ("wa" "th" "et") "rh" ("wa" "si" "ze")) ("anbi2i" ("w3a" ("wa" "ps" "ch") ("wa" "th" "et") ("wa" "si" "ze")) ("w3a" ("wa" "ps" "th") ("wa" "et" "si") ("wa" "ch" "ze")) ("w3a" "ph" "ta" "rh") ("_3bitri" ("w3a" ("wa" "ps" "ch") ("wa" "th" "et") ("wa" "si" "ze")) ("w3a" ("wa" "ps" "ch") ("wa" "th" "si") ("wa" "et" "ze")) ("w3a" ("wa" "ps" "th") ("wa" "ch" "si") ("wa" "et" "ze")) ("w3a" ("wa" "ps" "th") ("wa" "et" "si") ("wa" "ch" "ze")) ("_3bitr4i" ("wa" ("wa" "ps" "ch") ("wa" ("wa" "th" "et") ("wa" "si" "ze"))) ("wa" ("wa" "ps" "ch") ("wa" ("wa" "th" "si") ("wa" "et" "ze"))) ("w3a" ("wa" "ps" "ch") ("wa" "th" "et") ("wa" "si" "ze")) ("w3a" ("wa" "ps" "ch") ("wa" "th" "si") ("wa" "et" "ze")) ("anbi2i" ("wa" ("wa" "th" "et") ("wa" "si" "ze")) ("wa" ("wa" "th" "si") ("wa" "et" "ze")) ("wa" "ps" "ch") ("an4" "th" "et" "si" "ze")) ("_3anass" ("wa" "ps" "ch") ("wa" "th" "et") ("wa" "si" "ze")) ("_3anass" ("wa" "ps" "ch") ("wa" "th" "si") ("wa" "et" "ze"))) ("_3bitr4i" ("wa" ("wa" ("wa" "ps" "ch") ("wa" "th" "si")) ("wa" "et" "ze")) ("wa" ("wa" ("wa" "ps" "th") ("wa" "ch" "si")) ("wa" "et" "ze")) ("w3a" ("wa" "ps" "ch") ("wa" "th" "si") ("wa" "et" "ze")) ("w3a" ("wa" "ps" "th") ("wa" "ch" "si") ("wa" "et" "ze")) ("anbi1i" ("wa" ("wa" "ps" "ch") ("wa" "th" "si")) ("wa" ("wa" "ps" "th") ("wa" "ch" "si")) ("wa" "et" "ze") ("an4" "ps" "ch" "th" "si")) ("df_3an" ("wa" "ps" "ch") ("wa" "th" "si") ("wa" "et" "ze")) ("df_3an" ("wa" "ps" "th") ("wa" "ch" "si") ("wa" "et" "ze"))) ("_3bitr4i" ("wa" ("w3a" "ps" "ch" "et") ("w3a" "th" "si" "ze")) ("wa" ("w3a" "ps" "et" "ch") ("w3a" "th" "si" "ze")) ("w3a" ("wa" "ps" "th") ("wa" "ch" "si") ("wa" "et" "ze")) ("w3a" ("wa" "ps" "th") ("wa" "et" "si") ("wa" "ch" "ze")) ("anbi1i" ("w3a" "ps" "ch" "et") ("w3a" "ps" "et" "ch") ("w3a" "th" "si" "ze") ("_3ancomb" "ps" "ch" "et")) ("_3an6" "ps" "th" "ch" "si" "et" "ze") ("_3an6" "ps" "th" "et" "si" "ch" "ze")))))) ;; Extend wff definition to include alternative denial ('nand'). (term "wnan" ( ( "wff") ( "wff") ( "wff"))) ;; Define incompatibility, or alternative denial ('not-and' or 'nand'). This is also called the Sheffer stroke, represented by a vertical bar, but we use a different symbol to avoid ambiguity with other uses of the vertical bar. In the second edition of Principia Mathematica (1927), Russell and Whitehead used the Sheffer stroke and suggested it as a replacement for the "or" and "not" operations of the first edition. However, in practice, "or" and "not" are more widely used. After we define the constant true ` T. ` ( ~ df-tru ) and the constant false ` F. ` ( ~ df-fal ), we will be able to prove these truth table values: ` ( ( T. -/\ T. ) <-> F. ) ` ( ~ trunantru ), ` ( ( T. -/\ F. ) <-> T. ) ` ( ~ trunanfal ), ` ( ( F. -/\ T. ) <-> T. ) ` ( ~ falnantru ), and ` ( ( F. -/\ F. ) <-> T. ) ` ( ~ falnanfal ). Contrast with ` /\ ` ( ~ df-an ), ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), and ` \/_ ` ( ~ df-xor ) . (Contributed by Jeff Hoffman, 19-Nov-2007.) (axiom "df_nan" (!! ( ("ph" ( "wff")) ("ps" ( "wff"))) ("wb" ("wnan" "ph" "ps") ("wn" ("wa" "ph" "ps"))))) ;; Write 'and' in terms of 'nand'. (Contributed by Mario Carneiro, 9-May-2015.) (theorem "nanan" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wa" "ph" "ps") ("wn" ("wnan" "ph" "ps"))) ("con2bii" ("wnan" "ph" "ps") ("wa" "ph" "ps") ("df_nan" "ph" "ps"))) ;; The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 7-Mar-2020.) (theorem "nancom" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wnan" "ph" "ps") ("wnan" "ps" "ph")) ("bitr4i" ("wnan" "ph" "ps") ("wn" ("wa" "ps" "ph")) ("wnan" "ps" "ph") ("xchbinx" ("wnan" "ph" "ps") ("wa" "ph" "ps") ("wa" "ps" "ph") ("df_nan" "ph" "ps") ("ancom" "ph" "ps")) ("df_nan" "ps" "ph"))) ;; Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 7-Mar-2020.) (theorem "nannan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wnan" "ph" ("wnan" "ch" "ps")) ("wi" "ph" ("wa" "ch" "ps"))) ("_3bitr4ri" ("wi" "ph" ("wn" ("wnan" "ch" "ps"))) ("wn" ("wa" "ph" ("wnan" "ch" "ps"))) ("wi" "ph" ("wa" "ch" "ps")) ("wnan" "ph" ("wnan" "ch" "ps")) ("imnan" "ph" ("wnan" "ch" "ps")) ("imbi2i" ("wa" "ch" "ps") ("wn" ("wnan" "ch" "ps")) "ph" ("nanan" "ch" "ps")) ("df_nan" "ph" ("wnan" "ch" "ps")))) ;; Show equivalence between implication and the Nicod version. To derive ~ nic-dfim , apply ~ nanbi . (Contributed by Jeff Hoffman, 19-Nov-2007.) (theorem "nanim" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wi" "ph" "ps") ("wnan" "ph" ("wnan" "ps" "ps"))) ("bitr2i" ("wnan" "ph" ("wnan" "ps" "ps")) ("wi" "ph" ("wa" "ps" "ps")) ("wi" "ph" "ps") ("nannan" "ph" "ps" "ps") ("anidmdbi" "ph" "ps"))) ;; Show equivalence between negation and the Nicod version. To derive ~ nic-dfneg , apply ~ nanbi . (Contributed by Jeff Hoffman, 19-Nov-2007.) (theorem "nannot" (for ("ps" ( "wff"))) (for) (for) ("wb" ("wn" "ps") ("wnan" "ps" "ps")) ("bicomi" ("wnan" "ps" "ps") ("wn" "ps") ("xchbinx" ("wnan" "ps" "ps") ("wa" "ps" "ps") "ps" ("df_nan" "ps" "ps") ("anidm" "ps")))) ;; Show equivalence between the biconditional and the Nicod version. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2020.) (theorem "nanbi" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" "ph" "ps") ("wnan" ("wnan" "ph" "ps") ("wnan" ("wnan" "ph" "ph") ("wnan" "ps" "ps")))) ("bitr4i" ("wb" "ph" "ps") ("wi" ("wnan" "ph" "ps") ("wa" ("wnan" "ph" "ph") ("wnan" "ps" "ps"))) ("wnan" ("wnan" "ph" "ps") ("wnan" ("wnan" "ph" "ph") ("wnan" "ps" "ps"))) ("_3bitri" ("wb" "ph" "ps") ("wo" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps"))) ("wi" ("wn" ("wa" "ph" "ps")) ("wa" ("wn" "ph") ("wn" "ps"))) ("wi" ("wnan" "ph" "ps") ("wa" ("wnan" "ph" "ph") ("wnan" "ps" "ps"))) ("dfbi3" "ph" "ps") ("df_or" ("wa" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps"))) ("imbi12i" ("wn" ("wa" "ph" "ps")) ("wnan" "ph" "ps") ("wa" ("wn" "ph") ("wn" "ps")) ("wa" ("wnan" "ph" "ph") ("wnan" "ps" "ps")) ("bicomi" ("wnan" "ph" "ps") ("wn" ("wa" "ph" "ps")) ("df_nan" "ph" "ps")) ("anbi12i" ("wn" "ph") ("wnan" "ph" "ph") ("wn" "ps") ("wnan" "ps" "ps") ("nannot" "ph") ("nannot" "ps")))) ("nannan" ("wnan" "ph" "ps") ("wnan" "ps" "ps") ("wnan" "ph" "ph")))) ;; Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (theorem "nanbi1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wb" ("wnan" "ph" "ch") ("wnan" "ps" "ch"))) ("_3bitr4g" ("wb" "ph" "ps") ("wn" ("wa" "ph" "ch")) ("wn" ("wa" "ps" "ch")) ("wnan" "ph" "ch") ("wnan" "ps" "ch") ("notbid" ("wb" "ph" "ps") ("wa" "ph" "ch") ("wa" "ps" "ch") ("anbi1" "ph" "ps" "ch")) ("df_nan" "ph" "ch") ("df_nan" "ps" "ch"))) ;; Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (theorem "nanbi2" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wb" "ph" "ps") ("wb" ("wnan" "ch" "ph") ("wnan" "ch" "ps"))) ("_3bitr4g" ("wb" "ph" "ps") ("wnan" "ph" "ch") ("wnan" "ps" "ch") ("wnan" "ch" "ph") ("wnan" "ch" "ps") ("nanbi1" "ph" "ps" "ch") ("nancom" "ch" "ph") ("nancom" "ch" "ps"))) ;; Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.) (theorem "nanbi12" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for) (for) ("wi" ("wa" ("wb" "ph" "ps") ("wb" "ch" "th")) ("wb" ("wnan" "ph" "ch") ("wnan" "ps" "th"))) ("sylan9bb" ("wb" "ph" "ps") ("wnan" "ph" "ch") ("wnan" "ps" "ch") ("wb" "ch" "th") ("wnan" "ps" "th") ("nanbi1" "ph" "ps" "ch") ("nanbi2" "ch" "th" "ps"))) ;; Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (theorem "nanbi1i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("nanbii_1" ("wb" "ph" "ps"))) (for) ("wb" ("wnan" "ph" "ch") ("wnan" "ps" "ch")) ("ax_mp" ("wb" "ph" "ps") ("wb" ("wnan" "ph" "ch") ("wnan" "ps" "ch")) "nanbii_1" ("nanbi1" "ph" "ps" "ch"))) ;; Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (theorem "nanbi2i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for ("nanbii_1" ("wb" "ph" "ps"))) (for) ("wb" ("wnan" "ch" "ph") ("wnan" "ch" "ps")) ("ax_mp" ("wb" "ph" "ps") ("wb" ("wnan" "ch" "ph") ("wnan" "ch" "ps")) "nanbii_1" ("nanbi2" "ph" "ps" "ch"))) ;; Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.) (theorem "nanbi12i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("nanbii_1" ("wb" "ph" "ps")) ("nanbi12i_2" ("wb" "ch" "th"))) (for) ("wb" ("wnan" "ph" "ch") ("wnan" "ps" "th")) ("mp2an" ("wb" "ph" "ps") ("wb" "ch" "th") ("wb" ("wnan" "ph" "ch") ("wnan" "ps" "th")) "nanbii_1" "nanbi12i_2" ("nanbi12" "ph" "ps" "ch" "th"))) ;; Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (theorem "nanbi1d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("nanbid_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("wnan" "ps" "th") ("wnan" "ch" "th"))) ("syl" "ph" ("wb" "ps" "ch") ("wb" ("wnan" "ps" "th") ("wnan" "ch" "th")) "nanbid_1" ("nanbi1" "ps" "ch" "th"))) ;; Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (theorem "nanbi2d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("nanbid_1" ("wi" "ph" ("wb" "ps" "ch")))) (for) ("wi" "ph" ("wb" ("wnan" "th" "ps") ("wnan" "th" "ch"))) ("syl" "ph" ("wb" "ps" "ch") ("wb" ("wnan" "th" "ps") ("wnan" "th" "ch")) "nanbid_1" ("nanbi2" "ps" "ch" "th"))) ;; Join two logical equivalences with anti-conjunction. (Contributed by Scott Fenton, 2-Jan-2018.) (theorem "nanbi12d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("nanbid_1" ("wi" "ph" ("wb" "ps" "ch"))) ("nanbi12d_2" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" ("wnan" "ps" "th") ("wnan" "ch" "ta"))) ("syl2anc" "ph" ("wb" "ps" "ch") ("wb" "th" "ta") ("wb" ("wnan" "ps" "th") ("wnan" "ch" "ta")) "nanbid_1" "nanbi12d_2" ("nanbi12" "ps" "ch" "th" "ta"))) ;; Extend wff definition to include exclusive disjunction ('xor'). (term "wxo" ( ( "wff") ( "wff") ( "wff"))) ;; Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. After we define the constant true ` T. ` ( ~ df-tru ) and the constant false ` F. ` ( ~ df-fal ), we will be able to prove these truth table values: ` ( ( T. \/_ T. ) <-> F. ) ` ( ~ truxortru ), ` ( ( T. \/_ F. ) <-> T. ) ` ( ~ truxorfal ), ` ( ( F. \/_ T. ) <-> T. ) ` ( ~ falxortru ), and ` ( ( F. \/_ F. ) <-> F. ) ` ( ~ falxorfal ). Contrast with ` /\ ` ( ~ df-an ), ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), and ` -/\ ` ( ~ df-nan ) . (Contributed by FL, 22-Nov-2010.) (axiom "df_xor" (!! ( ("ph" ( "wff")) ("ps" ( "wff"))) ("wb" ("wxo" "ph" "ps") ("wn" ("wb" "ph" "ps"))))) ;; Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "xnor" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wb" "ph" "ps") ("wn" ("wxo" "ph" "ps"))) ("con2bii" ("wxo" "ph" "ps") ("wb" "ph" "ps") ("df_xor" "ph" "ps"))) ;; The connector ` \/_ ` is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "xorcom" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wxo" "ph" "ps") ("wxo" "ps" "ph")) ("_3bitr4i" ("wn" ("wb" "ph" "ps")) ("wn" ("wb" "ps" "ph")) ("wxo" "ph" "ps") ("wxo" "ps" "ph") ("notbii" ("wb" "ph" "ps") ("wb" "ps" "ph") ("bicom" "ph" "ps")) ("df_xor" "ph" "ps") ("df_xor" "ps" "ph"))) ;; The connector ` \/_ ` is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 20-Jun-2020.) (theorem "xorass" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wxo" ("wxo" "ph" "ps") "ch") ("wxo" "ph" ("wxo" "ps" "ch"))) ("_3bitr4i" ("wn" ("wb" ("wxo" "ph" "ps") "ch")) ("wn" ("wb" "ph" ("wxo" "ps" "ch"))) ("wxo" ("wxo" "ph" "ps") "ch") ("wxo" "ph" ("wxo" "ps" "ch")) ("_3bitr2ri" ("wn" ("wb" "ph" ("wxo" "ps" "ch"))) ("wb" "ph" ("wn" ("wxo" "ps" "ch"))) ("wb" ("wn" ("wxo" "ph" "ps")) "ch") ("wn" ("wb" ("wxo" "ph" "ps") "ch")) ("xor3" "ph" ("wxo" "ps" "ch")) ("_3bitr3i" ("wb" ("wb" "ph" "ps") "ch") ("wb" "ph" ("wb" "ps" "ch")) ("wb" ("wn" ("wxo" "ph" "ps")) "ch") ("wb" "ph" ("wn" ("wxo" "ps" "ch"))) ("biass" "ph" "ps" "ch") ("bibi1i" ("wb" "ph" "ps") ("wn" ("wxo" "ph" "ps")) "ch" ("xnor" "ph" "ps")) ("bibi2i" ("wb" "ps" "ch") ("wn" ("wxo" "ps" "ch")) "ph" ("xnor" "ps" "ch"))) ("nbbn" ("wxo" "ph" "ps") "ch")) ("df_xor" ("wxo" "ph" "ps") "ch") ("df_xor" "ph" ("wxo" "ps" "ch")))) ;; This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (theorem "excxor" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wxo" "ph" "ps") ("wo" ("wa" "ph" ("wn" "ps")) ("wa" ("wn" "ph") "ps"))) ("_3bitri" ("wxo" "ph" "ps") ("wn" ("wb" "ph" "ps")) ("wo" ("wa" "ph" ("wn" "ps")) ("wa" "ps" ("wn" "ph"))) ("wo" ("wa" "ph" ("wn" "ps")) ("wa" ("wn" "ph") "ps")) ("df_xor" "ph" "ps") ("xor" "ph" "ps") ("orbi2i" ("wa" "ps" ("wn" "ph")) ("wa" ("wn" "ph") "ps") ("wa" "ph" ("wn" "ps")) ("ancom" "ps" ("wn" "ph"))))) ;; Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "xor2" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wxo" "ph" "ps") ("wa" ("wo" "ph" "ps") ("wn" ("wa" "ph" "ps")))) ("bitri" ("wxo" "ph" "ps") ("wn" ("wb" "ph" "ps")) ("wa" ("wo" "ph" "ps") ("wn" ("wa" "ph" "ps"))) ("df_xor" "ph" "ps") ("nbi2" "ph" "ps"))) ;; XOR implies OR. (Contributed by BJ, 19-Apr-2019.) (theorem "xoror" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wxo" "ph" "ps") ("wo" "ph" "ps")) ("simplbi" ("wxo" "ph" "ps") ("wo" "ph" "ps") ("wn" ("wa" "ph" "ps")) ("xor2" "ph" "ps"))) ;; XOR implies NAND. (Contributed by BJ, 19-Apr-2019.) (theorem "xornan" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wxo" "ph" "ps") ("wn" ("wa" "ph" "ps"))) ("simprbi" ("wxo" "ph" "ps") ("wo" "ph" "ps") ("wn" ("wa" "ph" "ps")) ("xor2" "ph" "ps"))) ;; XOR implies NAND (written with the ` -/\ ` connector). (Contributed by BJ, 19-Apr-2019.) (theorem "xornan2" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wxo" "ph" "ps") ("wnan" "ph" "ps")) ("sylibr" ("wxo" "ph" "ps") ("wn" ("wa" "ph" "ps")) ("wnan" "ph" "ps") ("xornan" "ph" "ps") ("df_nan" "ph" "ps"))) ;; The connector ` \/_ ` is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.) (theorem "xorneg2" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wxo" "ph" ("wn" "ps")) ("wn" ("wxo" "ph" "ps"))) ("_3bitr2i" ("wxo" "ph" ("wn" "ps")) ("wn" ("wb" "ph" ("wn" "ps"))) ("wb" "ph" "ps") ("wn" ("wxo" "ph" "ps")) ("df_xor" "ph" ("wn" "ps")) ("pm5_18" "ph" "ps") ("xnor" "ph" "ps"))) ;; The connector ` \/_ ` is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.) (theorem "xorneg1" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wxo" ("wn" "ph") "ps") ("wn" ("wxo" "ph" "ps"))) ("bitri" ("wxo" ("wn" "ph") "ps") ("wxo" "ps" ("wn" "ph")) ("wn" ("wxo" "ph" "ps")) ("xorcom" ("wn" "ph") "ps") ("xchbinx" ("wxo" "ps" ("wn" "ph")) ("wxo" "ps" "ph") ("wxo" "ph" "ps") ("xorneg2" "ps" "ph") ("xorcom" "ps" "ph")))) ;; The connector ` \/_ ` is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "xorneg" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wb" ("wxo" ("wn" "ph") ("wn" "ps")) ("wxo" "ph" "ps")) ("bitr4i" ("wxo" ("wn" "ph") ("wn" "ps")) ("wn" ("wxo" "ph" ("wn" "ps"))) ("wxo" "ph" "ps") ("xorneg1" "ph" ("wn" "ps")) ("con2bii" ("wxo" "ph" ("wn" "ps")) ("wxo" "ph" "ps") ("xorneg2" "ph" "ps")))) ;; Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "xorbi12i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff"))) (for ("xorbi12_1" ("wb" "ph" "ps")) ("xorbi12_2" ("wb" "ch" "th"))) (for) ("wb" ("wxo" "ph" "ch") ("wxo" "ps" "th")) ("_3bitr4i" ("wn" ("wb" "ph" "ch")) ("wn" ("wb" "ps" "th")) ("wxo" "ph" "ch") ("wxo" "ps" "th") ("notbii" ("wb" "ph" "ch") ("wb" "ps" "th") ("bibi12i" "ph" "ps" "ch" "th" "xorbi12_1" "xorbi12_2")) ("df_xor" "ph" "ch") ("df_xor" "ps" "th"))) ;; Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "xorbi12d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for ("xor12d_1" ("wi" "ph" ("wb" "ps" "ch"))) ("xor12d_2" ("wi" "ph" ("wb" "th" "ta")))) (for) ("wi" "ph" ("wb" ("wxo" "ps" "th") ("wxo" "ch" "ta"))) ("_3bitr4g" "ph" ("wn" ("wb" "ps" "th")) ("wn" ("wb" "ch" "ta")) ("wxo" "ps" "th") ("wxo" "ch" "ta") ("notbid" "ph" ("wb" "ps" "th") ("wb" "ch" "ta") ("bibi12d" "ph" "ps" "ch" "th" "ta" "xor12d_1" "xor12d_2")) ("df_xor" "ps" "th") ("df_xor" "ch" "ta"))) ;; Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though ~ xordi does not necessarily hold, in part because the usual definition of xor is subtly different in intuitionistic logic. (Contributed by David A. Wheeler, 7-Oct-2018.) (theorem "anxordi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wa" "ph" ("wxo" "ps" "ch")) ("wxo" ("wa" "ph" "ps") ("wa" "ph" "ch"))) ("_3bitr4i" ("wa" "ph" ("wn" ("wb" "ps" "ch"))) ("wn" ("wb" ("wa" "ph" "ps") ("wa" "ph" "ch"))) ("wa" "ph" ("wxo" "ps" "ch")) ("wxo" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("xordi" "ph" "ps" "ch") ("anbi2i" ("wxo" "ps" "ch") ("wn" ("wb" "ps" "ch")) "ph" ("df_xor" "ps" "ch")) ("df_xor" ("wa" "ph" "ps") ("wa" "ph" "ch")))) ;; Exclusive-or variant of the law of the excluded middle ( ~ exmid ). This statement is ancient, going back to at least Stoic logic. This statement does not necessarily hold in intuitionistic logic. (Contributed by David A. Wheeler, 23-Feb-2019.) (theorem "xorexmid" (for ("ph" ( "wff"))) (for) (for) ("wxo" "ph" ("wn" "ph")) ("mpbir" ("wxo" "ph" ("wn" "ph")) ("wn" ("wb" "ph" ("wn" "ph"))) ("pm5_19" "ph") ("df_xor" "ph" ("wn" "ph")))) (sort "setvar") ;; Extend wff definition to include the universal quantifier ('for all'). ` A. x ph ` is read " ` ph ` (phi) is true for all ` x ` ." Typically, in its final application ` ph ` would be replaced with a wff containing a (free) occurrence of the variable ` x ` , for example ` x = y ` . In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of ` x ` . When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same. (term "wal" ( ( "setvar" "wff") ( "wff"))) (sort "class") ;; This syntax construction states that a variable ` x ` , which has been declared to be a setvar variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder ` { y | y e. x } ` is a class by ~ cab . Since (when ` y ` is distinct from ` x ` ) we have ` x = { y | y e. x } ` by ~ cvjust , we can argue that the syntax " ` class x ` " can be viewed as an abbreviation for " ` class { y | y e. x } ` ". See the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." While it is tempting and perhaps occasionally useful to view ~ cv as a "type conversion" from a setvar variable to a class variable, keep in mind that ~ cv is intrinsically no different from any other class-building syntax such as ~ cab , ~ cun , or ~ c0 . For a general discussion of the theory of classes and the role of ~ cv , see ~ mmset.html#class . (The description above applies to set theory, not predicate calculus. The purpose of introducing ` class x ` here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the ~ weq of predicate calculus from the ~ wceq of set theory, so that we don't overload the ` = ` connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.) (term "cv" ( ( "setvar") ( "class"))) ;; Extend wff definition to include class equality. For a general discussion of the theory of classes, see ~ mmset.html#class . (The purpose of introducing ` wff A = B ` here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the ~ weq of predicate calculus in terms of the ~ wceq of set theory, so that we don't "overload" the ` = ` connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the ` = ` in ` x = y ` could be the ` = ` of either ~ weq or ~ wceq , although mathematically it makes no difference. The class variables ` A ` and ` B ` are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See ~ df-cleq for more information on the set theory usage of ~ wceq .) (term "wceq" ( ( "class") ( "class") ( "wff"))) ;; The constant ` T. ` is a wff. (term "wtru" ( ( "wff"))) ;; Soundness justification theorem for ~ df-tru . (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) (theorem "trujust" (for) (for) (for) ("wb" ("wi" ("wal" (fn ("x2" "setvar") ("wceq" ("cv" "x2") ("cv" "x2")))) ("wal" (fn ("x2" "setvar") ("wceq" ("cv" "x2") ("cv" "x2"))))) ("wi" ("wal" (fn ("y" "setvar") ("wceq" ("cv" "y") ("cv" "y")))) ("wal" (fn ("y" "setvar") ("wceq" ("cv" "y") ("cv" "y")))))) ("_2th" ("wi" ("wal" (fn ("x2" "setvar") ("wceq" ("cv" "x2") ("cv" "x2")))) ("wal" (fn ("x2" "setvar") ("wceq" ("cv" "x2") ("cv" "x2"))))) ("wi" ("wal" (fn ("y" "setvar") ("wceq" ("cv" "y") ("cv" "y")))) ("wal" (fn ("y" "setvar") ("wceq" ("cv" "y") ("cv" "y"))))) ("id" ("wal" (fn ("x2" "setvar") ("wceq" ("cv" "x2") ("cv" "x2"))))) ("id" ("wal" (fn ("y" "setvar") ("wceq" ("cv" "y") ("cv" "y"))))))) ;; Definition of the truth value "true", or "verum", denoted by ` T. ` . This is a tautology, as proved by ~ tru . In this definition, an instance of ~ id is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular ~ id instance was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by ~ tru , and other proofs should depend on ~ tru (directly or indirectly) instead of this definition, since there are many alternate ways to define ` T. ` . (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) Use ~ tru instead. (New usage is discouraged.) (axiom "df_tru" (!! () ("wb" "wtru" ("wi" ("wal" (fn ("x2" "setvar") ("wceq" ("cv" "x2") ("cv" "x2")))) ("wal" (fn ("x2" "setvar") ("wceq" ("cv" "x2") ("cv" "x2")))))))) ;; The truth value ` T. ` is provable. (Contributed by Anthony Hart, 13-Oct-2010.) (theorem "tru" (for) (for) (for) "wtru" ("mpbir" "wtru" ("wi" ("wal" (fn ("x2" "setvar") ("wceq" ("cv" "x2") ("cv" "x2")))) ("wal" (fn ("x2" "setvar") ("wceq" ("cv" "x2") ("cv" "x2"))))) ("id" ("wal" (fn ("x2" "setvar") ("wceq" ("cv" "x2") ("cv" "x2"))))) "df_tru")) ;; The constant ` F. ` is a wff. (term "wfal" ( ( "wff"))) ;; Definition of the truth value "false", or "falsum", denoted by ` F. ` . See also ~ df-tru . (Contributed by Anthony Hart, 22-Oct-2010.) (axiom "df_fal" (!! () ("wb" "wfal" ("wn" "wtru")))) ;; The truth value ` F. ` is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) (theorem "fal" (for) (for) (for) ("wn" "wfal") ("mtbir" "wfal" ("wn" "wtru") ("notnoti" "wtru" "tru") "df_fal")) ;; An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.) (theorem "dftru2" (for ("ph" ( "wff"))) (for) (for) ("wb" "wtru" ("wi" "ph" "ph")) ("_2th" "wtru" ("wi" "ph" "ph") "tru" ("id" "ph"))) ;; A proposition is equivalent to it being implied by ` T. ` . Closed form of ~ trud . Dual of ~ dfnot . It is to ~ tbtru what ~ a1bi is to ~ tbt . (Contributed by BJ, 26-Oct-2019.) (theorem "trut" (for ("ph" ( "wff"))) (for) (for) ("wb" "ph" ("wi" "wtru" "ph")) ("a1bi" "wtru" "ph" "tru")) ;; Eliminate ` T. ` as an antecedent. A proposition implied by ` T. ` is true. (Contributed by Mario Carneiro, 13-Mar-2014.) (theorem "trud" (for ("ph" ( "wff"))) (for ("trud_1" ("wi" "wtru" "ph"))) (for) "ph" ("ax_mp" "wtru" "ph" "tru" "trud_1")) ;; A proposition is equivalent to itself being equivalent to ` T. ` . (Contributed by Anthony Hart, 14-Aug-2011.) (theorem "tbtru" (for ("ph" ( "wff"))) (for) (for) ("wb" "ph" ("wb" "ph" "wtru")) ("tbt" "wtru" "ph" "tru")) ;; The negation of a proposition is equivalent to itself being equivalent to ` F. ` . (Contributed by Anthony Hart, 14-Aug-2011.) (theorem "nbfal" (for ("ph" ( "wff"))) (for) (for) ("wb" ("wn" "ph") ("wb" "ph" "wfal")) ("nbn" "wfal" "ph" "fal")) ;; A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) (theorem "bitru" (for ("ph" ( "wff"))) (for ("bitru_1" "ph")) (for) ("wb" "ph" "wtru") ("_2th" "ph" "wtru" "bitru_1" "tru")) ;; A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) (theorem "bifal" (for ("ph" ( "wff"))) (for ("bifal_1" ("wn" "ph"))) (for) ("wb" "ph" "wfal") ("_2false" "ph" "wfal" "bifal_1" "fal")) ;; The truth value ` F. ` implies anything. Also called the "principle of explosion", or "ex falso [[sequitur]] quodlibet" (Latin for "from falsehood, anything [[follows]]"). (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) (theorem "falim" (for ("ph" ( "wff"))) (for) (for) ("wi" "wfal" "ph") ("pm2_21i" "wfal" "ph" "fal")) ;; The truth value ` F. ` implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.) (theorem "falimd" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "wfal") "ps") ("adantl" "wfal" "ps" "ph" ("falim" "ps"))) ;; Anything implies ` T. ` . (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) (theorem "a1tru" (for ("ph" ( "wff"))) (for) (for) ("wi" "ph" "wtru") ("a1i" "wtru" "ph" "tru")) ;; True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) (theorem "truan" (for ("ph" ( "wff"))) (for) (for) ("wb" ("wa" "wtru" "ph") "ph") ("bicomi" "ph" ("wa" "wtru" "ph") ("biantrur" "wtru" "ph" "tru"))) ;; Given falsum ` F. ` , we can define the negation of a wff ` ph ` as the statement that ` F. ` follows from assuming ` ph ` . (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) (theorem "dfnot" (for ("ph" ( "wff"))) (for) (for) ("wb" ("wn" "ph") ("wi" "ph" "wfal")) ("ax_mp" ("wn" "wfal") ("wb" ("wn" "ph") ("wi" "ph" "wfal")) "fal" ("mtt" "wfal" "ph"))) ;; Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) (theorem "inegd" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("inegd_1" ("wi" ("wa" "ph" "ps") "wfal"))) (for) ("wi" "ph" ("wn" "ps")) ("sylibr" "ph" ("wi" "ps" "wfal") ("wn" "ps") ("ex" "ph" "ps" "wfal" "inegd_1") ("dfnot" "ps"))) ;; Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.) (theorem "efald" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("efald_1" ("wi" ("wa" "ph" ("wn" "ps")) "wfal"))) (for) ("wi" "ph" "ps") ("notnotrd" "ph" "ps" ("inegd" "ph" ("wn" "ps") "efald_1"))) ;; If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.) (theorem "pm2_21fal" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for ("pm2_21fal_1" ("wi" "ph" "ps")) ("pm2_21fal_2" ("wi" "ph" ("wn" "ps")))) (for) ("wi" "ph" "wfal") ("pm2_21dd" "ph" "ps" "wfal" "pm2_21fal_1" "pm2_21fal_2")) ;; A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (theorem "truantru" (for) (for) (for) ("wb" ("wa" "wtru" "wtru") "wtru") ("anidm" "wtru")) ;; A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (theorem "truanfal" (for) (for) (for) ("wb" ("wa" "wtru" "wfal") "wfal") ("truan" "wfal")) ;; A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (theorem "falantru" (for) (for) (for) ("wb" ("wa" "wfal" "wtru") "wfal") ("bifal" ("wa" "wfal" "wtru") ("intnanr" "wfal" "wtru" "fal"))) ;; A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (theorem "falanfal" (for) (for) (for) ("wb" ("wa" "wfal" "wfal") "wfal") ("anidm" "wfal")) ;; A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "truortru" (for) (for) (for) ("wb" ("wo" "wtru" "wtru") "wtru") ("oridm" "wtru")) ;; A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (theorem "truorfal" (for) (for) (for) ("wb" ("wo" "wtru" "wfal") "wtru") ("bitru" ("wo" "wtru" "wfal") ("orci" "wtru" "wfal" "tru"))) ;; A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (theorem "falortru" (for) (for) (for) ("wb" ("wo" "wfal" "wtru") "wtru") ("bitru" ("wo" "wfal" "wtru") ("olci" "wtru" "wfal" "tru"))) ;; A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "falorfal" (for) (for) (for) ("wb" ("wo" "wfal" "wfal") "wfal") ("oridm" "wfal")) ;; A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (theorem "truimtru" (for) (for) (for) ("wb" ("wi" "wtru" "wtru") "wtru") ("bitru" ("wi" "wtru" "wtru") ("id" "wtru"))) ;; A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "truimfal" (for) (for) (for) ("wb" ("wi" "wtru" "wfal") "wfal") ("bicomi" "wfal" ("wi" "wtru" "wfal") ("trut" "wfal"))) ;; A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (theorem "falimtru" (for) (for) (for) ("wb" ("wi" "wfal" "wtru") "wtru") ("bitru" ("wi" "wfal" "wtru") ("falim" "wtru"))) ;; A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (theorem "falimfal" (for) (for) (for) ("wb" ("wi" "wfal" "wfal") "wtru") ("bitru" ("wi" "wfal" "wfal") ("id" "wfal"))) ;; A ` -. ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (theorem "nottru" (for) (for) (for) ("wb" ("wn" "wtru") "wfal") ("bicomi" "wfal" ("wn" "wtru") "df_fal")) ;; A ` -. ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "notfal" (for) (for) (for) ("wb" ("wn" "wfal") "wtru") ("bitru" ("wn" "wfal") "fal")) ;; A ` <-> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "trubitru" (for) (for) (for) ("wb" ("wb" "wtru" "wtru") "wtru") ("bitru" ("wb" "wtru" "wtru") ("biid" "wtru"))) ;; A ` <-> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.) (theorem "falbitru" (for) (for) (for) ("wb" ("wb" "wfal" "wtru") "wfal") ("bicomi" "wfal" ("wb" "wfal" "wtru") ("tbtru" "wfal"))) ;; A ` <-> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.) (theorem "trubifal" (for) (for) (for) ("wb" ("wb" "wtru" "wfal") "wfal") ("bitri" ("wb" "wtru" "wfal") ("wb" "wfal" "wtru") "wfal" ("bicom" "wtru" "wfal") "falbitru")) ;; A ` <-> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "falbifal" (for) (for) (for) ("wb" ("wb" "wfal" "wfal") "wtru") ("bitru" ("wb" "wfal" "wfal") ("biid" "wfal"))) ;; A ` -/\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "trunantru" (for) (for) (for) ("wb" ("wnan" "wtru" "wtru") "wfal") ("bitr3i" ("wnan" "wtru" "wtru") ("wn" "wtru") "wfal" ("nannot" "wtru") "nottru")) ;; A ` -/\ ` identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.) (theorem "trunanfal" (for) (for) (for) ("wb" ("wnan" "wtru" "wfal") "wtru") ("bitri" ("wnan" "wtru" "wfal") ("wn" "wfal") "wtru" ("xchbinx" ("wnan" "wtru" "wfal") ("wa" "wtru" "wfal") "wfal" ("df_nan" "wtru" "wfal") "truanfal") "notfal")) ;; A ` -/\ ` identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "falnantru" (for) (for) (for) ("wb" ("wnan" "wfal" "wtru") "wtru") ("bitri" ("wnan" "wfal" "wtru") ("wnan" "wtru" "wfal") "wtru" ("nancom" "wfal" "wtru") "trunanfal")) ;; A ` -/\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (theorem "falnanfal" (for) (for) (for) ("wb" ("wnan" "wfal" "wfal") "wtru") ("bitr3i" ("wnan" "wfal" "wfal") ("wn" "wfal") "wtru" ("nannot" "wfal") "notfal")) ;; A ` \/_ ` identity. (Contributed by David A. Wheeler, 8-May-2015.) (theorem "truxortru" (for) (for) (for) ("wb" ("wxo" "wtru" "wtru") "wfal") ("bitri" ("wxo" "wtru" "wtru") ("wn" "wtru") "wfal" ("xchbinx" ("wxo" "wtru" "wtru") ("wb" "wtru" "wtru") "wtru" ("df_xor" "wtru" "wtru") "trubitru") "nottru")) ;; A ` \/_ ` identity. (Contributed by David A. Wheeler, 8-May-2015.) (theorem "truxorfal" (for) (for) (for) ("wb" ("wxo" "wtru" "wfal") "wtru") ("bitri" ("wxo" "wtru" "wfal") ("wn" "wfal") "wtru" ("xchbinx" ("wxo" "wtru" "wfal") ("wb" "wtru" "wfal") "wfal" ("df_xor" "wtru" "wfal") "trubifal") "notfal")) ;; A ` \/_ ` identity. (Contributed by David A. Wheeler, 9-May-2015.) (Proof shortened by Wolf Lammen, 10-Jul-2020.) (theorem "falxortru" (for) (for) (for) ("wb" ("wxo" "wfal" "wtru") "wtru") ("bitri" ("wxo" "wfal" "wtru") ("wxo" "wtru" "wfal") "wtru" ("xorcom" "wfal" "wtru") "truxorfal")) ;; A ` \/_ ` identity. (Contributed by David A. Wheeler, 9-May-2015.) (theorem "falxorfal" (for) (for) (for) ("wb" ("wxo" "wfal" "wfal") "wfal") ("bitri" ("wxo" "wfal" "wfal") ("wn" "wtru") "wfal" ("xchbinx" ("wxo" "wfal" "wfal") ("wb" "wfal" "wfal") "wtru" ("df_xor" "wfal" "wfal") "falbifal") "nottru")) ;; Syntax for the "sum" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.) (term "whad" ( ( "wff") ( "wff") ( "wff") ( "wff"))) ;; Definition of the "sum" output of the full adder (triple exclusive disjunction, or XOR3). (Contributed by Mario Carneiro, 4-Sep-2016.) (axiom "df_had" (!! ( ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) ("wb" ("whad" "ph" "ps" "ch") ("wxo" ("wxo" "ph" "ps") "ch")))) ;; Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "hadbi123d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("hadbid_1" ("wi" "ph" ("wb" "ps" "ch"))) ("hadbid_2" ("wi" "ph" ("wb" "th" "ta"))) ("hadbid_3" ("wi" "ph" ("wb" "et" "ze")))) (for) ("wi" "ph" ("wb" ("whad" "ps" "th" "et") ("whad" "ch" "ta" "ze"))) ("_3bitr4g" "ph" ("wxo" ("wxo" "ps" "th") "et") ("wxo" ("wxo" "ch" "ta") "ze") ("whad" "ps" "th" "et") ("whad" "ch" "ta" "ze") ("xorbi12d" "ph" ("wxo" "ps" "th") ("wxo" "ch" "ta") "et" "ze" ("xorbi12d" "ph" "ps" "ch" "th" "ta" "hadbid_1" "hadbid_2") "hadbid_3") ("df_had" "ps" "th" "et") ("df_had" "ch" "ta" "ze"))) ;; Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "hadbi123i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("hadbii_1" ("wb" "ph" "ps")) ("hadbii_2" ("wb" "ch" "th")) ("hadbii_3" ("wb" "ta" "et"))) (for) ("wb" ("whad" "ph" "ch" "ta") ("whad" "ps" "th" "et")) ("trud" ("wb" ("whad" "ph" "ch" "ta") ("whad" "ps" "th" "et")) ("hadbi123d" "wtru" "ph" "ps" "ch" "th" "ta" "et" ("a1i" ("wb" "ph" "ps") "wtru" "hadbii_1") ("a1i" ("wb" "ch" "th") "wtru" "hadbii_2") ("a1i" ("wb" "ta" "et") "wtru" "hadbii_3")))) ;; Associative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "hadass" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("whad" "ph" "ps" "ch") ("wxo" "ph" ("wxo" "ps" "ch"))) ("bitri" ("whad" "ph" "ps" "ch") ("wxo" ("wxo" "ph" "ps") "ch") ("wxo" "ph" ("wxo" "ps" "ch")) ("df_had" "ph" "ps" "ch") ("xorass" "ph" "ps" "ch"))) ;; The adder sum is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "hadbi" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("whad" "ph" "ps" "ch") ("wb" ("wb" "ph" "ps") "ch")) ("_3bitr4i" ("wxo" ("wxo" "ph" "ps") "ch") ("wn" ("wb" ("wxo" "ph" "ps") "ch")) ("whad" "ph" "ps" "ch") ("wb" ("wb" "ph" "ps") "ch") ("df_xor" ("wxo" "ph" "ps") "ch") ("df_had" "ph" "ps" "ch") ("bitri" ("wb" ("wb" "ph" "ps") "ch") ("wb" ("wn" ("wxo" "ph" "ps")) "ch") ("wn" ("wb" ("wxo" "ph" "ps") "ch")) ("bibi1i" ("wb" "ph" "ps") ("wn" ("wxo" "ph" "ps")) "ch" ("xnor" "ph" "ps")) ("nbbn" ("wxo" "ph" "ps") "ch")))) ;; Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "hadcoma" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("whad" "ph" "ps" "ch") ("whad" "ps" "ph" "ch")) ("_3bitr4i" ("wxo" ("wxo" "ph" "ps") "ch") ("wxo" ("wxo" "ps" "ph") "ch") ("whad" "ph" "ps" "ch") ("whad" "ps" "ph" "ch") ("xorbi12i" ("wxo" "ph" "ps") ("wxo" "ps" "ph") "ch" "ch" ("xorcom" "ph" "ps") ("biid" "ch")) ("df_had" "ph" "ps" "ch") ("df_had" "ps" "ph" "ch"))) ;; Commutative law for the adders sum. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "hadcomb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("whad" "ph" "ps" "ch") ("whad" "ph" "ch" "ps")) ("_3bitr4i" ("wxo" "ph" ("wxo" "ps" "ch")) ("wxo" "ph" ("wxo" "ch" "ps")) ("whad" "ph" "ps" "ch") ("whad" "ph" "ch" "ps") ("xorbi12i" "ph" "ph" ("wxo" "ps" "ch") ("wxo" "ch" "ps") ("biid" "ph") ("xorcom" "ps" "ch")) ("hadass" "ph" "ps" "ch") ("hadass" "ph" "ch" "ps"))) ;; Rotation law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "hadrot" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("whad" "ph" "ps" "ch") ("whad" "ps" "ch" "ph")) ("bitri" ("whad" "ph" "ps" "ch") ("whad" "ps" "ph" "ch") ("whad" "ps" "ch" "ph") ("hadcoma" "ph" "ps" "ch") ("hadcomb" "ps" "ph" "ch"))) ;; The adder sum distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) (theorem "hadnot" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wn" ("whad" "ph" "ps" "ch")) ("whad" ("wn" "ph") ("wn" "ps") ("wn" "ch"))) ("_3bitr4i" ("wb" ("wb" "ph" "ps") ("wn" "ch")) ("wb" ("wb" ("wn" "ph") ("wn" "ps")) ("wn" "ch")) ("wn" ("whad" "ph" "ps" "ch")) ("whad" ("wn" "ph") ("wn" "ps") ("wn" "ch")) ("bibi1i" ("wb" "ph" "ps") ("wb" ("wn" "ph") ("wn" "ps")) ("wn" "ch") ("notbi" "ph" "ps")) ("xchnxbir" ("wb" ("wb" "ph" "ps") "ch") ("wb" ("wb" "ph" "ps") ("wn" "ch")) ("whad" "ph" "ps" "ch") ("xor3" ("wb" "ph" "ps") "ch") ("hadbi" "ph" "ps" "ch")) ("hadbi" ("wn" "ph") ("wn" "ps") ("wn" "ch")))) ;; If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) (theorem "had1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" "ph" ("wb" ("whad" "ph" "ps" "ch") ("wb" "ps" "ch"))) ("biimpri" ("wb" ("whad" "ph" "ps" "ch") ("wb" "ps" "ch")) "ph" ("mpbir" ("wb" ("wb" ("whad" "ph" "ps" "ch") ("wb" "ps" "ch")) "ph") ("wb" ("whad" "ph" "ps" "ch") ("wb" ("wb" "ps" "ch") "ph")) ("bitri" ("whad" "ph" "ps" "ch") ("whad" "ps" "ch" "ph") ("wb" ("wb" "ps" "ch") "ph") ("hadrot" "ph" "ps" "ch") ("hadbi" "ps" "ch" "ph")) ("biass" ("whad" "ph" "ps" "ch") ("wb" "ps" "ch") "ph")))) ;; If the first input is false, then the adder sum is equivalent to the exclusive disjunction of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jul-2020.) (theorem "had0" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wn" "ph") ("wb" ("whad" "ph" "ps" "ch") ("wxo" "ps" "ch"))) ("con4bid" ("wn" "ph") ("whad" "ph" "ps" "ch") ("wxo" "ps" "ch") ("_3bitr4g" ("wn" "ph") ("whad" ("wn" "ph") ("wn" "ps") ("wn" "ch")) ("wb" ("wn" "ps") ("wn" "ch")) ("wn" ("whad" "ph" "ps" "ch")) ("wn" ("wxo" "ps" "ch")) ("had1" ("wn" "ph") ("wn" "ps") ("wn" "ch")) ("hadnot" "ph" "ps" "ch") ("bitr3i" ("wn" ("wxo" "ps" "ch")) ("wb" "ps" "ch") ("wb" ("wn" "ps") ("wn" "ch")) ("xnor" "ps" "ch") ("notbi" "ps" "ch"))))) ;; The value of the adder sum is, if the first input is true, the biconditionality, and if the first input is false, the exclusive disjunction, of the other two inputs. (Contributed by BJ, 11-Aug-2020.) (theorem "hadifp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("whad" "ph" "ps" "ch") ("wif" "ph" ("wb" "ps" "ch") ("wxo" "ps" "ch"))) ("casesifp" "ph" ("whad" "ph" "ps" "ch") ("wb" "ps" "ch") ("wxo" "ps" "ch") ("had1" "ph" "ps" "ch") ("had0" "ph" "ps" "ch"))) ;; Syntax for the "carry" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.) (term "wcad" ( ( "wff") ( "wff") ( "wff") ( "wff"))) ;; Definition of the "carry" output of the full adder. It is true when at least two arguments are true, so it is equal to the "majority" function on three variables. See ~ cador and ~ cadan for alternate definitions. (Contributed by Mario Carneiro, 4-Sep-2016.) (axiom "df_cad" (!! ( ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) ("wb" ("wcad" "ph" "ps" "ch") ("wo" ("wa" "ph" "ps") ("wa" "ch" ("wxo" "ph" "ps")))))) ;; The adder carry in disjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) (theorem "cador" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wcad" "ph" "ps" "ch") ("w3o" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ps" "ch"))) ("_3bitr4i" ("wo" ("wa" "ph" "ps") ("wa" "ch" ("wxo" "ph" "ps"))) ("wo" ("wa" "ph" "ps") ("wo" ("wa" "ph" "ch") ("wa" "ps" "ch"))) ("wcad" "ph" "ps" "ch") ("w3o" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ps" "ch")) ("_3bitr4i" ("wi" ("wn" ("wa" "ph" "ps")) ("wa" "ch" ("wxo" "ph" "ps"))) ("wi" ("wn" ("wa" "ph" "ps")) ("wo" ("wa" "ph" "ch") ("wa" "ps" "ch"))) ("wo" ("wa" "ph" "ps") ("wa" "ch" ("wxo" "ph" "ps"))) ("wo" ("wa" "ph" "ps") ("wo" ("wa" "ph" "ch") ("wa" "ps" "ch"))) ("pm5_74i" ("wn" ("wa" "ph" "ps")) ("wa" "ch" ("wxo" "ph" "ps")) ("wo" ("wa" "ph" "ch") ("wa" "ps" "ch")) ("_3bitr3g" ("wn" ("wa" "ph" "ps")) ("wa" ("wxo" "ph" "ps") "ch") ("wa" ("wo" "ph" "ps") "ch") ("wa" "ch" ("wxo" "ph" "ps")) ("wo" ("wa" "ph" "ch") ("wa" "ps" "ch")) ("anbi1d" ("wn" ("wa" "ph" "ps")) ("wxo" "ph" "ps") ("wo" "ph" "ps") "ch" ("rbaib" ("wxo" "ph" "ps") ("wo" "ph" "ps") ("wn" ("wa" "ph" "ps")) ("xor2" "ph" "ps"))) ("ancom" ("wxo" "ph" "ps") "ch") ("andir" "ph" "ps" "ch"))) ("df_or" ("wa" "ph" "ps") ("wa" "ch" ("wxo" "ph" "ps"))) ("df_or" ("wa" "ph" "ps") ("wo" ("wa" "ph" "ch") ("wa" "ps" "ch")))) ("df_cad" "ph" "ps" "ch") ("_3orass" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ps" "ch")))) ;; The adder carry in conjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 25-Sep-2018.) (theorem "cadan" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wcad" "ph" "ps" "ch") ("w3a" ("wo" "ph" "ps") ("wo" "ph" "ch") ("wo" "ps" "ch"))) ("bitr4i" ("wcad" "ph" "ps" "ch") ("wa" ("wa" ("wo" "ph" "ps") ("wo" "ph" "ch")) ("wo" "ps" "ch")) ("w3a" ("wo" "ph" "ps") ("wo" "ph" "ch") ("wo" "ps" "ch")) ("_3bitri" ("wcad" "ph" "ps" "ch") ("wo" ("wa" "ph" ("wo" "ps" "ch")) ("wa" "ps" "ch")) ("wa" ("wo" "ph" ("wa" "ps" "ch")) ("wo" ("wo" "ps" "ch") ("wa" "ps" "ch"))) ("wa" ("wa" ("wo" "ph" "ps") ("wo" "ph" "ch")) ("wo" "ps" "ch")) ("_3bitr4i" ("w3o" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ps" "ch")) ("wo" ("wo" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("wa" "ps" "ch")) ("wcad" "ph" "ps" "ch") ("wo" ("wa" "ph" ("wo" "ps" "ch")) ("wa" "ps" "ch")) ("df_3or" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ps" "ch")) ("cador" "ph" "ps" "ch") ("orbi1i" ("wa" "ph" ("wo" "ps" "ch")) ("wo" ("wa" "ph" "ps") ("wa" "ph" "ch")) ("wa" "ps" "ch") ("andi" "ph" "ps" "ch"))) ("ordir" "ph" ("wo" "ps" "ch") ("wa" "ps" "ch")) ("anbi12i" ("wo" "ph" ("wa" "ps" "ch")) ("wa" ("wo" "ph" "ps") ("wo" "ph" "ch")) ("wo" ("wo" "ps" "ch") ("wa" "ps" "ch")) ("wo" "ps" "ch") ("ordi" "ph" "ps" "ch") ("bitr4i" ("wo" ("wo" "ps" "ch") ("wa" "ps" "ch")) ("wo" ("wa" "ps" "ch") ("wo" "ps" "ch")) ("wo" "ps" "ch") ("orcom" ("wo" "ps" "ch") ("wa" "ps" "ch")) ("mpbi" ("wi" ("wa" "ps" "ch") ("wo" "ps" "ch")) ("wb" ("wo" "ps" "ch") ("wo" ("wa" "ps" "ch") ("wo" "ps" "ch"))) ("animorl" "ps" "ch" "ch") ("pm4_72" ("wa" "ps" "ch") ("wo" "ps" "ch")))))) ("df_3an" ("wo" "ph" "ps") ("wo" "ph" "ch") ("wo" "ps" "ch")))) ;; Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "cadbi123d" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff")) ("ze" ( "wff"))) (for ("cadbid_1" ("wi" "ph" ("wb" "ps" "ch"))) ("cadbid_2" ("wi" "ph" ("wb" "th" "ta"))) ("cadbid_3" ("wi" "ph" ("wb" "et" "ze")))) (for) ("wi" "ph" ("wb" ("wcad" "ps" "th" "et") ("wcad" "ch" "ta" "ze"))) ("_3bitr4g" "ph" ("wo" ("wa" "ps" "th") ("wa" "et" ("wxo" "ps" "th"))) ("wo" ("wa" "ch" "ta") ("wa" "ze" ("wxo" "ch" "ta"))) ("wcad" "ps" "th" "et") ("wcad" "ch" "ta" "ze") ("orbi12d" "ph" ("wa" "ps" "th") ("wa" "ch" "ta") ("wa" "et" ("wxo" "ps" "th")) ("wa" "ze" ("wxo" "ch" "ta")) ("anbi12d" "ph" "ps" "ch" "th" "ta" "cadbid_1" "cadbid_2") ("anbi12d" "ph" "et" "ze" ("wxo" "ps" "th") ("wxo" "ch" "ta") "cadbid_3" ("xorbi12d" "ph" "ps" "ch" "th" "ta" "cadbid_1" "cadbid_2"))) ("df_cad" "ps" "th" "et") ("df_cad" "ch" "ta" "ze"))) ;; Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "cadbi123i" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff")) ("et" ( "wff"))) (for ("cadbii_1" ("wb" "ph" "ps")) ("cadbii_2" ("wb" "ch" "th")) ("cadbii_3" ("wb" "ta" "et"))) (for) ("wb" ("wcad" "ph" "ch" "ta") ("wcad" "ps" "th" "et")) ("trud" ("wb" ("wcad" "ph" "ch" "ta") ("wcad" "ps" "th" "et")) ("cadbi123d" "wtru" "ph" "ps" "ch" "th" "ta" "et" ("a1i" ("wb" "ph" "ps") "wtru" "cadbii_1") ("a1i" ("wb" "ch" "th") "wtru" "cadbii_2") ("a1i" ("wb" "ta" "et") "wtru" "cadbii_3")))) ;; Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "cadcoma" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wcad" "ph" "ps" "ch") ("wcad" "ps" "ph" "ch")) ("_3bitr4i" ("wo" ("wa" "ph" "ps") ("wa" "ch" ("wxo" "ph" "ps"))) ("wo" ("wa" "ps" "ph") ("wa" "ch" ("wxo" "ps" "ph"))) ("wcad" "ph" "ps" "ch") ("wcad" "ps" "ph" "ch") ("orbi12i" ("wa" "ph" "ps") ("wa" "ps" "ph") ("wa" "ch" ("wxo" "ph" "ps")) ("wa" "ch" ("wxo" "ps" "ph")) ("ancom" "ph" "ps") ("anbi2i" ("wxo" "ph" "ps") ("wxo" "ps" "ph") "ch" ("xorcom" "ph" "ps"))) ("df_cad" "ph" "ps" "ch") ("df_cad" "ps" "ph" "ch"))) ;; Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) (theorem "cadcomb" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wcad" "ph" "ps" "ch") ("wcad" "ph" "ch" "ps")) ("bitr4i" ("wcad" "ph" "ps" "ch") ("w3a" ("wo" "ph" "ch") ("wo" "ph" "ps") ("wo" "ch" "ps")) ("wcad" "ph" "ch" "ps") ("_3bitri" ("wcad" "ph" "ps" "ch") ("w3a" ("wo" "ph" "ps") ("wo" "ph" "ch") ("wo" "ps" "ch")) ("w3a" ("wo" "ph" "ch") ("wo" "ph" "ps") ("wo" "ps" "ch")) ("w3a" ("wo" "ph" "ch") ("wo" "ph" "ps") ("wo" "ch" "ps")) ("cadan" "ph" "ps" "ch") ("_3ancoma" ("wo" "ph" "ps") ("wo" "ph" "ch") ("wo" "ps" "ch")) ("_3anbi3i" ("wo" "ps" "ch") ("wo" "ch" "ps") ("wo" "ph" "ch") ("wo" "ph" "ps") ("orcom" "ps" "ch"))) ("cadan" "ph" "ch" "ps"))) ;; Rotation law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "cadrot" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wcad" "ph" "ps" "ch") ("wcad" "ps" "ch" "ph")) ("bitri" ("wcad" "ph" "ps" "ch") ("wcad" "ps" "ph" "ch") ("wcad" "ps" "ch" "ph") ("cadcoma" "ph" "ps" "ch") ("cadcomb" "ps" "ph" "ch"))) ;; The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) (theorem "cadnot" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wn" ("wcad" "ph" "ps" "ch")) ("wcad" ("wn" "ph") ("wn" "ps") ("wn" "ch"))) ("_3bitr4i" ("w3a" ("wn" ("wa" "ph" "ps")) ("wn" ("wa" "ph" "ch")) ("wn" ("wa" "ps" "ch"))) ("w3a" ("wo" ("wn" "ph") ("wn" "ps")) ("wo" ("wn" "ph") ("wn" "ch")) ("wo" ("wn" "ps") ("wn" "ch"))) ("wn" ("wcad" "ph" "ps" "ch")) ("wcad" ("wn" "ph") ("wn" "ps") ("wn" "ch")) ("_3anbi123i" ("wn" ("wa" "ph" "ps")) ("wo" ("wn" "ph") ("wn" "ps")) ("wn" ("wa" "ph" "ch")) ("wo" ("wn" "ph") ("wn" "ch")) ("wn" ("wa" "ps" "ch")) ("wo" ("wn" "ps") ("wn" "ch")) ("ianor" "ph" "ps") ("ianor" "ph" "ch") ("ianor" "ps" "ch")) ("xchnxbir" ("w3o" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ps" "ch")) ("w3a" ("wn" ("wa" "ph" "ps")) ("wn" ("wa" "ph" "ch")) ("wn" ("wa" "ps" "ch"))) ("wcad" "ph" "ps" "ch") ("_3ioran" ("wa" "ph" "ps") ("wa" "ph" "ch") ("wa" "ps" "ch")) ("cador" "ph" "ps" "ch")) ("cadan" ("wn" "ph") ("wn" "ps") ("wn" "ch")))) ;; If one input is true, then the adder carry is true exactly when at least one of the other two inputs is true. (Contributed by Mario Carneiro, 8-Sep-2016.) (Proof shortened by Wolf Lammen, 19-Jun-2020.) (theorem "cad1" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" "ch" ("wb" ("wcad" "ph" "ps" "ch") ("wo" "ph" "ps"))) ("syl6rbbr" "ch" ("wo" "ph" "ps") ("wa" ("wo" "ph" "ps") ("wa" ("wo" "ph" "ch") ("wo" "ps" "ch"))) ("wcad" "ph" "ps" "ch") ("biantrud" "ch" ("wa" ("wo" "ph" "ch") ("wo" "ps" "ch")) ("wo" "ph" "ps") ("jca" "ch" ("wo" "ph" "ch") ("wo" "ps" "ch") ("olc" "ch" "ph") ("olc" "ch" "ps"))) ("bitri" ("wcad" "ph" "ps" "ch") ("w3a" ("wo" "ph" "ps") ("wo" "ph" "ch") ("wo" "ps" "ch")) ("wa" ("wo" "ph" "ps") ("wa" ("wo" "ph" "ch") ("wo" "ps" "ch"))) ("cadan" "ph" "ps" "ch") ("_3anass" ("wo" "ph" "ps") ("wo" "ph" "ch") ("wo" "ps" "ch"))))) ;; If one input is false, then the adder carry is true exactly when both of the other two inputs are true. (Contributed by Mario Carneiro, 8-Sep-2016.) (theorem "cad0" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wn" "ch") ("wb" ("wcad" "ph" "ps" "ch") ("wa" "ph" "ps"))) ("syl5bb" ("wcad" "ph" "ps" "ch") ("wo" ("wa" "ph" "ps") ("wa" "ch" ("wxo" "ph" "ps"))) ("wn" "ch") ("wa" "ph" "ps") ("df_cad" "ph" "ps" "ch") ("impbid1" ("wn" "ch") ("wo" ("wa" "ph" "ps") ("wa" "ch" ("wxo" "ph" "ps"))) ("wa" "ph" "ps") ("jaod" ("wn" "ch") ("wa" "ph" "ps") ("wa" "ph" "ps") ("wa" "ch" ("wxo" "ph" "ps")) ("idd" ("wn" "ch") ("wa" "ph" "ps")) ("adantrd" ("wn" "ch") "ch" ("wa" "ph" "ps") ("wxo" "ph" "ps") ("pm2_21" "ch" ("wa" "ph" "ps")))) ("orc" ("wa" "ph" "ps") ("wa" "ch" ("wxo" "ph" "ps")))))) ;; The value of the carry is, if the input carry is true, the disjunction, and if the input carry is false, the conjunction, of the other two inputs. (Contributed by BJ, 8-Oct-2019.) (theorem "cadifp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wb" ("wcad" "ph" "ps" "ch") ("wif" "ch" ("wo" "ph" "ps") ("wa" "ph" "ps"))) ("casesifp" "ch" ("wcad" "ph" "ps" "ch") ("wo" "ph" "ps") ("wa" "ph" "ps") ("cad1" "ph" "ps" "ch") ("cad0" "ph" "ps" "ch"))) ;; If (at least) two inputs are true, then the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "cad11" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wa" "ph" "ps") ("wcad" "ph" "ps" "ch")) ("sylibr" ("wa" "ph" "ps") ("wo" ("wa" "ph" "ps") ("wa" "ch" ("wxo" "ph" "ps"))) ("wcad" "ph" "ps" "ch") ("orc" ("wa" "ph" "ps") ("wa" "ch" ("wxo" "ph" "ps"))) ("df_cad" "ph" "ps" "ch"))) ;; The adder carry is true as soon as its first two inputs are the truth constant. (Contributed by Mario Carneiro, 4-Sep-2016.) (theorem "cadtru" (for ("ph" ( "wff"))) (for) (for) ("wcad" "wtru" "wtru" "ph") ("mp2an" "wtru" "wtru" ("wcad" "wtru" "wtru" "ph") "tru" "tru" ("cad11" "wtru" "wtru" "ph"))) ;; A single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. (Contributed by BJ, 4-Apr-2021.) (theorem "minimp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff")) ("th" ( "wff")) ("ta" ( "wff"))) (for) (for) ("wi" "ph" ("wi" ("wi" "ps" "ch") ("wi" ("wi" ("wi" "th" "ps") ("wi" "ch" "ta")) ("wi" "ps" "ta")))) ("a1i" ("wi" ("wi" "ps" "ch") ("wi" ("wi" ("wi" "th" "ps") ("wi" "ch" "ta")) ("wi" "ps" "ta"))) "ph" ("com12" ("wi" ("wi" "th" "ps") ("wi" "ch" "ta")) ("wi" "ps" "ch") ("wi" "ps" "ta") ("a2d" ("wi" ("wi" "th" "ps") ("wi" "ch" "ta")) "ps" "ch" "ta" ("jarr" "th" "ps" ("wi" "ch" "ta")))))) ;; Derivation of sylsimp ( ~ jarr ) from ~ ax-mp and ~ minimp . (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "minimp_sylsimp" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" "ps" "ch")) ("mp2b" ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")))) ("wi" ("wi" "ph" "ps") "ch")) ("wi" ("wi" "ph" "ps") "ch")) ("wi" ("wi" "ph" "ps") ("wi" ("wi" ("wi" "ph" "ps") "ch") "ch")) ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" "ps" "ch")) ("mp2b" ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")))) ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))))) ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))))) ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")))) ("wi" ("wi" "ph" "ps") "ch"))) ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")))) ("wi" ("wi" "ph" "ps") "ch"))) ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")))) ("wi" ("wi" "ph" "ps") "ch")) ("wi" ("wi" "ph" "ps") "ch")) ("minimp" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")))) ("wi" "ph" "ps") ("wi" "ph" "ps") ("wi" "ph" "ps") ("wi" "ph" "ps")) ("ax_mp" ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")))) ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")))) ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))))) ("wi" ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")) ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ps") "ch") ("wi" "ph" "ps")) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps"))) ("wi" ("wi" "ph" "ps") ("wi" "ph" "ps")))) ("wi" ("wi" ("wi" 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(Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "minimp_ax1" (for ("ph" ( "wff")) ("ps" ( "wff"))) (for) (for) ("wi" "ph" ("wi" "ps" "ph")) ("ax_mp" ("wi" ("wi" ("wi" "ph" "ps") "ph") ("wi" "ps" "ph")) ("wi" "ph" ("wi" "ps" "ph")) ("minimp_sylsimp" "ph" "ps" "ph") ("minimp_sylsimp" ("wi" "ph" "ps") "ph" ("wi" "ps" "ph")))) ;; Derivation of a commuted form of ~ ax-2 from ~ ax-mp and ~ minimp . (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) (theorem "minimp_ax2c" (for ("ph" ( "wff")) ("ps" ( "wff")) ("ch" ( "wff"))) (for) (for) ("wi" ("wi" "ph" "ps") ("wi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ph" "ch"))) ("mp2" ("wi" ("wi" "ph" "ps") ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ph") ("wi" ("wi" ("wi" "ph" "ph") ("wi" "ph" "ph")) ("wi" "ph" "ph"))) "ph") ("wi" "ps" "ch")) ("wi" "ph" "ch"))) ("wi" ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ph") ("wi" ("wi" ("wi" "ph" "ph") ("wi" "ph" "ph")) ("wi" "ph" "ph"))) "ph") ("wi" "ps" "ch")) ("wi" "ph" "ch")) ("wi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ph" "ch"))) ("wi" ("wi" "ph" "ps") ("wi" ("wi" "ph" ("wi" "ps" "ch")) ("wi" "ph" "ch"))) ("ax_mp" ("wi" "ph" ("wi" ("wi" "ph" "ph") ("wi" ("wi" ("wi" "ph" "ph") ("wi" "ph" "ph")) ("wi" "ph" "ph")))) ("wi" ("wi" "ph" "ps") ("wi" ("wi" ("wi" ("wi" ("wi" "ph" "ph") ("wi" ("wi" ("wi" "ph" "ph") ("wi" "ph" "ph")) ("wi" "ph" "ph"))) "ph") ("wi" "ps" "ch")) ("wi" "ph" "ch"))) ("minimp" "ph" "ph" "ph" "ph" "ph") ("minimp" ("wi" "ph" ("wi" ("wi" "ph" "ph") ("wi" 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