(sort wff provable)

--| (_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.)
(pub theorem a1ii ((ph wff ()) (ps wff ()))
  ((a1ii_1 ph)
    (a1ii_2 ps))
  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.)
(pub theorem idi ((ph wff ()))
  ((idi_1 ph))
  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 ((ph 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 ((ph wff ()) (ps 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.)
(pub theorem mp2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mp2_1 ph)
    (mp2_2 ps)
    (mp2_3 (wi ph (wi ps ch))))
  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.)
(pub theorem mp2b ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mp2b_1 ph)
    (mp2b_2 (wi ph ps))
    (mp2b_3 (wi ps ch)))
  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.)
(pub theorem a1i ((ph wff ()) (ps wff ()))
  ((a1i_1 ph))
  (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.)
(pub theorem _2a1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_2a1i_1 ph))
  (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.)
(pub theorem mp1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mp1i_1 ph)
    (mp1i_2 (wi ph ps)))
  (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.)
(pub theorem a2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((a2i_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem mpd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpd_1 (wi ph ps))
    (mpd_2 (wi ph (wi ps ch))))
  (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.)
(pub theorem imim2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((imim2i_1 (wi ph ps)))
  (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.)
(pub theorem syl ((ph wff ()) (ps wff ()) (ch wff ()))
  ((syl_1 (wi ph ps))
    (syl_2 (wi ps ch)))
  (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.)
(pub theorem _3syl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3syl_1 (wi ph ps))
    (_3syl_2 (wi ps ch))
    (_3syl_3 (wi ch th)))
  (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.)
(pub theorem _4syl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_4syl_1 (wi ph ps))
    (_4syl_2 (wi ps ch))
    (_4syl_3 (wi ch th))
    (_4syl_4 (wi th ta)))
  (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.)
(pub theorem mpi ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpi_1 ps)
    (mpi_2 (wi ph (wi ps ch))))
  (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.)
(pub theorem mpisyl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpisyl_1 (wi ph ps))
    (mpisyl_2 ch)
    (mpisyl_3 (wi ps (wi ch th))))
  (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.)
(pub theorem id ((ph wff ()))
  ()
  (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.)
(pub theorem idALT ((ph wff ()))
  ()
  (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.)
(pub theorem idd ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem a1d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((a1d_1 (wi ph ps)))
  (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.)
(pub theorem _2a1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_2a1d_1 (wi ph ps)))
  (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.)
(pub theorem a1i13 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((a1i13_1 (wi ps th)))
  (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.)
(pub theorem _2a1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem a2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((a2d_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem sylcom ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylcom_1 (wi ph (wi ps ch)))
    (sylcom_2 (wi ps (wi ch th))))
  (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.)
(pub theorem syl5com ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5com_1 (wi ph ps))
    (syl5com_2 (wi ch (wi ps th))))
  (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.)
(pub theorem com12 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((com12_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem syl11 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl11_1 (wi ph (wi ps ch)))
    (syl11_2 (wi th ph)))
  (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.)
(pub theorem syl5 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5_1 (wi ph ps))
    (syl5_2 (wi ch (wi ps th))))
  (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.)
(pub theorem syl6 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl6_1 (wi ph (wi ps ch)))
    (syl6_2 (wi ch th)))
  (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.)
(pub theorem syl56 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl56_1 (wi ph ps))
    (syl56_2 (wi ch (wi ps th)))
    (syl56_3 (wi th ta)))
  (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.)
(pub theorem syl6com ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl6com_1 (wi ph (wi ps ch)))
    (syl6com_2 (wi ch th)))
  (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.)
(pub theorem mpcom ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpcom_1 (wi ps ph))
    (mpcom_2 (wi ph (wi ps ch))))
  (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.)
(pub theorem syli ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syli_1 (wi ps (wi ph ch)))
    (syli_2 (wi ch (wi ph th))))
  (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.)
(pub theorem syl2im ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl2im_1 (wi ph ps))
    (syl2im_2 (wi ch th))
    (syl2im_3 (wi ps (wi th ta))))
  (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.)
(pub theorem syl2imc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl2im_1 (wi ph ps))
    (syl2im_2 (wi ch th))
    (syl2im_3 (wi ps (wi th ta))))
  (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.)
(pub theorem pm2_27 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem mpdd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpdd_1 (wi ph (wi ps ch)))
    (mpdd_2 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem mpid ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpid_1 (wi ph ch))
    (mpid_2 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem mpdi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpdi_1 (wi ps ch))
    (mpdi_2 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem mpii ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpii_1 ch)
    (mpii_2 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem syld ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syld_1 (wi ph (wi ps ch)))
    (syld_2 (wi ph (wi ch th))))
  (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.)
(pub theorem syldc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syld_1 (wi ph (wi ps ch)))
    (syld_2 (wi ph (wi ch th))))
  (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.)
(pub theorem mp2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mp2d_1 (wi ph ps))
    (mp2d_2 (wi ph ch))
    (mp2d_3 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem a1dd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((a1dd_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem _2a1dd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_2a1dd_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem pm2_43i ((ph wff ()) (ps wff ()))
  ((pm2_43i_1 (wi ph (wi ph ps))))
  (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.)
(pub theorem pm2_43d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_43d_1 (wi ph (wi ps (wi ps ch)))))
  (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.)
(pub theorem pm2_43a ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_43a_1 (wi ps (wi ph (wi ps ch)))))
  (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.)
(pub theorem pm2_43b ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_43b_1 (wi ps (wi ph (wi ps ch)))))
  (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.)
(pub theorem pm2_43 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem imim2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((imim2d_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem imim2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem embantd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((embantd_1 (wi ph ps))
    (embantd_2 (wi ph (wi ch th))))
  (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.)
(pub theorem _3syld ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3syld_1 (wi ph (wi ps ch)))
    (_3syld_2 (wi ph (wi ch th)))
    (_3syld_3 (wi ph (wi th ta))))
  (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.)
(pub theorem sylsyld ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylsyld_1 (wi ph ps))
    (sylsyld_2 (wi ph (wi ch th)))
    (sylsyld_3 (wi ps (wi th ta))))
  (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.)
(pub theorem imim12i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((imim12i_1 (wi ph ps))
    (imim12i_2 (wi ch th)))
  (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.)
(pub theorem imim1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((imim1i_1 (wi ph ps)))
  (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.)
(pub theorem imim3i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((imim3i_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem sylc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylc_1 (wi ph ps))
    (sylc_2 (wi ph ch))
    (sylc_3 (wi ps (wi ch th))))
  (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.)
(pub theorem syl3c ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl3c_1 (wi ph ps))
    (syl3c_2 (wi ph ch))
    (syl3c_3 (wi ph th))
    (syl3c_4 (wi ps (wi ch (wi th ta)))))
  (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.)
(pub theorem syl6mpi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl6mpi_1 (wi ph (wi ps ch)))
    (syl6mpi_2 th)
    (syl6mpi_3 (wi ch (wi th ta))))
  (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.)
(pub theorem mpsyl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpsyl_1 ph)
    (mpsyl_2 (wi ps ch))
    (mpsyl_3 (wi ph (wi ch th))))
  (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.)
(pub theorem mpsylsyld ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mpsylsyld_1 ph)
    (mpsylsyld_2 (wi ps (wi ch th)))
    (mpsylsyld_3 (wi ph (wi th ta))))
  (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.)
(pub theorem syl6c ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl6c_1 (wi ph (wi ps ch)))
    (syl6c_2 (wi ph (wi ps th)))
    (syl6c_3 (wi ch (wi th ta))))
  (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.)
(pub theorem syl6ci ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl6ci_1 (wi ph (wi ps ch)))
    (syl6ci_2 (wi ph th))
    (syl6ci_3 (wi ch (wi th ta))))
  (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.)
(pub theorem syldd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syldd_1 (wi ph (wi ps (wi ch th))))
    (syldd_2 (wi ph (wi ps (wi th ta)))))
  (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.)
(pub theorem syl5d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl5d_1 (wi ph (wi ps ch)))
    (syl5d_2 (wi ph (wi th (wi ch ta)))))
  (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.)
(pub theorem syl7 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl7_1 (wi ph ps))
    (syl7_2 (wi ch (wi th (wi ps ta)))))
  (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.)
(pub theorem syl6d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl6d_1 (wi ph (wi ps (wi ch th))))
    (syl6d_2 (wi ph (wi th ta))))
  (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.)
(pub theorem syl8 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl8_1 (wi ph (wi ps (wi ch th))))
    (syl8_2 (wi th ta)))
  (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.)
(pub theorem syl9 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl9_1 (wi ph (wi ps ch)))
    (syl9_2 (wi th (wi ch ta))))
  (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.)
(pub theorem syl9r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl9r_1 (wi ph (wi ps ch)))
    (syl9r_2 (wi th (wi ch ta))))
  (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.)
(pub theorem syl10 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((syl10_1 (wi ph (wi ps ch)))
    (syl10_2 (wi ph (wi ps (wi th ta))))
    (syl10_3 (wi ch (wi ta et))))
  (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.)
(pub theorem a1ddd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((a1ddd_1 (wi ph (wi ps (wi ch ta)))))
  (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.)
(pub theorem imim12d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imim12d_1 (wi ph (wi ps ch)))
    (imim12d_2 (wi ph (wi th ta))))
  (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.)
(pub theorem imim1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((imim1d_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem imim1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_83 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem peirceroll ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem com23 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((com3_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem com3r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((com3_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem com13 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((com3_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem com3l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((com3_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem pm2_04 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem com34 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((com4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem com4l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((com4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem com4t ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((com4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem com4r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((com4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem com24 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((com4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem com14 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((com4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem com45 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((com5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem com35 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((com5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem com25 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((com5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem com5l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((com5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem com15 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((com5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem com52l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((com5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem com52r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((com5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem com5r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((com5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem imim12 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem jarr ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_86d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((pm2_86d_1 (wi ph (wi (wi ps ch) (wi ps th)))))
  (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.)
(pub theorem pm2_86 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_86i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_86i_1 (wi (wi ph ps) (wi ph ch))))
  (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.)
(pub theorem loolin ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem loowoz ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem con4 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem con4i ((ph wff ()) (ps wff ()))
  ((con4i_1 (wi (wn ph) (wn ps))))
  (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.)
(pub theorem con4d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((con4d_1 (wi ph (wi (wn ps) (wn ch)))))
  (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.)
(pub theorem mt4 ((ph wff ()) (ps wff ()))
  ((mt4_1 ph)
    (mt4_2 (wi (wn ps) (wn ph))))
  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.)
(pub theorem pm2_21i ((ph wff ()) (ps wff ()))
  ((pm2_21i_1 (wn ph)))
  (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.)
(pub theorem pm2_24ii ((ph wff ()) (ps wff ()))
  ((pm2_24ii_1 ph)
    (pm2_24ii_2 (wn ph)))
  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.)
(pub theorem pm2_21d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_21d_1 (wi ph (wn ps))))
  (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.)
(pub theorem pm2_21ddALT ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_21ddALT_1 (wi ph ps))
    (pm2_21ddALT_2 (wi ph (wn ps))))
  (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.)
(pub theorem pm2_21 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_24 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_18 ((ph wff ()))
  ()
  (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.)
(pub theorem pm2_18i ((ph wff ()))
  ((pm2_18i_1 (wi (wn ph) ph)))
  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.)
(pub theorem pm2_18d ((ph wff ()) (ps wff ()))
  ((pm2_18d_1 (wi ph (wi (wn ps) ps))))
  (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.)
(pub theorem notnotr ((ph wff ()))
  ()
  (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.)
(pub theorem notnotri ((ph wff ()))
  ((notnotri_1 (wn (wn ph))))
  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.)
(pub theorem notnotriOLD ((ph wff ()))
  ((notnotri_1 (wn (wn ph))))
  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.)
(pub theorem notnotrd ((ph wff ()) (ps wff ()))
  ((notnotrd_1 (wi ph (wn (wn ps)))))
  (wi ph ps)
  ()
  (syl (ph (wn (wn ps)) ps)
    notnotrd_1
    (notnotr (ps))))

--| A contraposition deduction. (Contributed by NM, 19-Aug-1993.)
(pub theorem con2d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((con2d_1 (wi ph (wi ps (wn ch)))))
  (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.)
(pub theorem con2 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem mt2d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mt2d_1 (wi ph ch))
    (mt2d_2 (wi ph (wi ps (wn ch)))))
  (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.)
(pub theorem mt2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mt2i_1 ch)
    (mt2i_2 (wi ph (wi ps (wn ch)))))
  (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.)
(pub theorem nsyl3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((nsyl3_1 (wi ph (wn ps)))
    (nsyl3_2 (wi ch ps)))
  (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.)
(pub theorem con2i ((ph wff ()) (ps wff ()))
  ((con2i_a (wi ph (wn ps))))
  (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.)
(pub theorem nsyl ((ph wff ()) (ps wff ()) (ch wff ()))
  ((nsyl_1 (wi ph (wn ps)))
    (nsyl_2 (wi ch ps)))
  (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.)
(pub theorem notnot ((ph wff ()))
  ()
  (wi ph (wn (wn ph)))
  ()
  (con2i ((wn ph) ph)
    (id ((wn ph)))))

--| Inference associated with ~ notnot . (Contributed by NM, 27-Feb-2008.)
(pub theorem notnoti ((ph wff ()))
  ((notnoti_1 ph))
  (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.)
(pub theorem notnotd ((ph wff ()) (ps wff ()))
  ((notnotd_1 (wi ph ps)))
  (wi ph (wn (wn ps)))
  ()
  (syl (ph ps (wn (wn ps)))
    notnotd_1
    (notnot (ps))))

--| A contraposition deduction. (Contributed by NM, 27-Dec-1992.)
(pub theorem con1d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((con1d_1 (wi ph (wi (wn ps) ch))))
  (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.)
(pub theorem mt3d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mt3d_1 (wi ph (wn ch)))
    (mt3d_2 (wi ph (wi (wn ps) ch))))
  (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.)
(pub theorem mt3i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mt3i_1 (wn ch))
    (mt3i_2 (wi ph (wi (wn ps) ch))))
  (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.)
(pub theorem nsyl2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((nsyl2_1 (wi ph (wn ps)))
    (nsyl2_2 (wi (wn ch) ps)))
  (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.)
(pub theorem con1 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem con1i ((ph wff ()) (ps wff ()))
  ((con1i_1 (wi (wn ph) ps)))
  (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.)
(pub theorem con4iOLD ((ph wff ()) (ps wff ()))
  ((con4iOLD_1 (wi (wn ph) (wn ps))))
  (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.)
(pub theorem pm2_24i ((ph wff ()) (ps wff ()))
  ((pm2_24i_1 ph))
  (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.)
(pub theorem pm2_24d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_24d_1 (wi ph ps)))
  (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.)
(pub theorem con3d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((con3d_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem con3 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem con3i ((ph wff ()) (ps wff ()))
  ((con3i_a (wi ph ps)))
  (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.)
(pub theorem con3rr3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((con3rr3_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem mt4d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mt4d_1 (wi ph ps))
    (mt4d_2 (wi ph (wi (wn ch) (wn ps)))))
  (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.)
(pub theorem mt4i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mt4i_1 ch)
    (mt4i_2 (wi ph (wi (wn ps) (wn ch)))))
  (wi ph ps)
  ()
  (mt4d (ph ch ps)
    (a1i (ch ph)
      mt4i_1)
    mt4i_2))

--| A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
(pub theorem nsyld ((ph wff ()) (ps wff ()) (ch wff ()) (ta wff ()))
  ((nsyld_1 (wi ph (wi ps (wn ch))))
    (nsyld_2 (wi ph (wi ta ch))))
  (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.)
(pub theorem nsyli ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((nsyli_1 (wi ph (wi ps ch)))
    (nsyli_2 (wi th (wn ch))))
  (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.)
(pub theorem nsyl4 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((nsyl4_1 (wi ph ps))
    (nsyl4_2 (wi (wn ph) ch)))
  (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.)
(pub theorem pm3_2im ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem mth8 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem jc ((ph wff ()) (ps wff ()) (ch wff ()))
  ((jc_1 (wi ph ps))
    (jc_2 (wi ph ch)))
  (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.)
(pub theorem impi ((ph wff ()) (ps wff ()) (ch wff ()))
  ((impi_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem expi ((ph wff ()) (ps wff ()) (ch wff ()))
  ((expi_1 (wi (wn (wi ph (wn ps))) ch)))
  (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.)
(pub theorem simprim ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem simplim ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_5 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_51 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_521 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_52 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem expt ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem impt ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_61d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_61d_1 (wi ph (wi ps ch)))
    (pm2_61d_2 (wi ph (wi (wn ps) ch))))
  (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.)
(pub theorem pm2_61d1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_61d1_1 (wi ph (wi ps ch)))
    (pm2_61d1_2 (wi (wn ps) ch)))
  (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.)
(pub theorem pm2_61d2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_61d2_1 (wi ph (wi (wn ps) ch)))
    (pm2_61d2_2 (wi ps ch)))
  (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.)
(pub theorem ja ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ja_1 (wi (wn ph) ch))
    (ja_2 (wi ps ch)))
  (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.)
(pub theorem jad ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jad_1 (wi ph (wi (wn ps) th)))
    (jad_2 (wi ph (wi ch th))))
  (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.)
(pub theorem jarl ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_61i ((ph wff ()) (ps wff ()))
  ((pm2_61i_1 (wi ph ps))
    (pm2_61i_2 (wi (wn ph) ps)))
  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.)
(pub theorem pm2_61ii ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_61ii_1 (wi (wn ph) (wi (wn ps) ch)))
    (pm2_61ii_2 (wi ph ch))
    (pm2_61ii_3 (wi ps ch)))
  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.)
(pub theorem pm2_61nii ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_61nii_1 (wi ph (wi ps ch)))
    (pm2_61nii_2 (wi (wn ph) ch))
    (pm2_61nii_3 (wi (wn ps) ch)))
  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.)
(pub theorem pm2_61iii ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((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)))
  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.)
(pub theorem pm2_01 ((ph wff ()))
  ()
  (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.)
(pub theorem pm2_01d ((ph wff ()) (ps wff ()))
  ((pm2_01d_1 (wi ph (wi ps (wn ps)))))
  (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.)
(pub theorem pm2_6 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_61 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_65 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_65i ((ph wff ()) (ps wff ()))
  ((pm2_65i_1 (wi ph ps))
    (pm2_65i_2 (wi ph (wn ps))))
  (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.)
(pub theorem pm2_21dd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_21dd_1 (wi ph ps))
    (pm2_21dd_2 (wi ph (wn ps))))
  (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.)
(pub theorem pm2_65d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_65d_1 (wi ph (wi ps ch)))
    (pm2_65d_2 (wi ph (wi ps (wn ch)))))
  (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.)
(pub theorem mto ((ph wff ()) (ps wff ()))
  ((mto_1 (wn ps))
    (mto_2 (wi ph ps)))
  (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.)
(pub theorem mtod ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mtod_1 (wi ph (wn ch)))
    (mtod_2 (wi ph (wi ps ch))))
  (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.)
(pub theorem mtoi ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mtoi_1 (wn ch))
    (mtoi_2 (wi ph (wi ps ch))))
  (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.)
(pub theorem mt2 ((ph wff ()) (ps wff ()))
  ((mt2_1 ps)
    (mt2_2 (wi ph (wn ps))))
  (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.)
(pub theorem mt3 ((ph wff ()) (ps wff ()))
  ((mt3_1 (wn ps))
    (mt3_2 (wi (wn ph) ps)))
  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.)
(pub theorem peirce ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem looinv ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem bijust ((ph wff ()) (ps wff ()))
  ()
  (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 ((ph wff ()) (ps 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.)
(pub theorem impbi ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem impbii ((ph wff ()) (ps wff ()))
  ((impbii_1 (wi ph ps))
    (impbii_2 (wi ps ph)))
  (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.)
(pub theorem impbidd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((impbidd_1 (wi ph (wi ps (wi ch th))))
    (impbidd_2 (wi ph (wi ps (wi th ch)))))
  (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.)
(pub theorem impbid21d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((impbid21d_1 (wi ps (wi ch th)))
    (impbid21d_2 (wi ph (wi th ch))))
  (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.)
(pub theorem impbid ((ph wff ()) (ps wff ()) (ch wff ()))
  ((impbid_1 (wi ph (wi ps ch)))
    (impbid_2 (wi ph (wi ch ps))))
  (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.)
(pub theorem dfbi1 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem dfbi1ALT ((ph wff ()) (ps wff ()))
  ()
  (wb (wb ph ps) (wn (wi (wi ph ps) (wn (wi ps ph)))))
  ((ch wff) (th wff))
  (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))
    (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.)
(pub theorem biimp ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem biimpi ((ph wff ()) (ps wff ()))
  ((biimpi_1 (wb ph ps)))
  (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.)
(pub theorem sylbi ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylbi_1 (wb ph ps))
    (sylbi_2 (wi ps ch)))
  (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.)
(pub theorem sylib ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylib_1 (wi ph ps))
    (sylib_2 (wb ps ch)))
  (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.)
(pub theorem sylbb ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylbb_1 (wb ph ps))
    (sylbb_2 (wb ps ch)))
  (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.)
(pub theorem biimpr ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem bicom1 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem bicom ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem bicomd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((bicomd_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem bicomi ((ph wff ()) (ps wff ()))
  ((bicomi_1 (wb ph ps)))
  (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.)
(pub theorem impbid1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((impbid1_1 (wi ph (wi ps ch)))
    (impbid1_2 (wi ch ps)))
  (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.)
(pub theorem impbid2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((impbid2_1 (wi ps ch))
    (impbid2_2 (wi ph (wi ch ps))))
  (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.)
(pub theorem impcon4bid ((ph wff ()) (ps wff ()) (ch wff ()))
  ((impcon4bid_1 (wi ph (wi ps ch)))
    (impcon4bid_2 (wi ph (wi (wn ps) (wn ch)))))
  (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.)
(pub theorem biimpri ((ph wff ()) (ps wff ()))
  ((biimpri_1 (wb ph ps)))
  (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.)
(pub theorem biimpd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((biimpd_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem mpbi ((ph wff ()) (ps wff ()))
  ((mpbi_min ph)
    (mpbi_maj (wb ph ps)))
  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.)
(pub theorem mpbir ((ph wff ()) (ps wff ()))
  ((mpbir_min ps)
    (mpbir_maj (wb ph ps)))
  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.)
(pub theorem mpbid ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpbid_min (wi ph ps))
    (mpbid_maj (wi ph (wb ps ch))))
  (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.)
(pub theorem mpbii ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpbii_min ps)
    (mpbii_maj (wi ph (wb ps ch))))
  (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.)
(pub theorem sylibr ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylibr_1 (wi ph ps))
    (sylibr_2 (wb ch ps)))
  (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.)
(pub theorem sylbir ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylbir_1 (wb ps ph))
    (sylbir_2 (wi ps ch)))
  (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.)
(pub theorem sylbbr ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylbbr_1 (wb ph ps))
    (sylbbr_2 (wb ps ch)))
  (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.)
(pub theorem sylbb1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylbb1_1 (wb ph ps))
    (sylbb1_2 (wb ph ch)))
  (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.)
(pub theorem sylbb2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylbb2_1 (wb ph ps))
    (sylbb2_2 (wb ch ps)))
  (wi ph ch)
  ()
  (sylbi (ph ps ch)
    sylbb2_1
    (biimpri (ch ps)
      sylbb2_2)))

--| A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(pub theorem sylibd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylibd_1 (wi ph (wi ps ch)))
    (sylibd_2 (wi ph (wb ch th))))
  (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.)
(pub theorem sylbid ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylbid_1 (wi ph (wb ps ch)))
    (sylbid_2 (wi ph (wi ch th))))
  (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.)
(pub theorem mpbidi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpbidi_min (wi th (wi ph ps)))
    (mpbidi_maj (wi ph (wb ps ch))))
  (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.)
(pub theorem syl5bi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5bi_1 (wb ph ps))
    (syl5bi_2 (wi ch (wi ps th))))
  (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.)
(pub theorem syl5bir ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5bir_1 (wb ps ph))
    (syl5bir_2 (wi ch (wi ps th))))
  (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.)
(pub theorem syl5ib ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5ib_1 (wi ph ps))
    (syl5ib_2 (wi ch (wb ps th))))
  (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.)
(pub theorem syl5ibcom ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5ib_1 (wi ph ps))
    (syl5ib_2 (wi ch (wb ps th))))
  (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.)
(pub theorem syl5ibr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5ibr_1 (wi ph th))
    (syl5ibr_2 (wi ch (wb ps th))))
  (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.)
(pub theorem syl5ibrcom ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5ibr_1 (wi ph th))
    (syl5ibr_2 (wi ch (wb ps th))))
  (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.)
(pub theorem biimprd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((biimprd_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem biimpcd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((biimpcd_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem biimprcd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((biimpcd_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem syl6ib ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl6ib_1 (wi ph (wi ps ch)))
    (syl6ib_2 (wb ch th)))
  (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.)
(pub theorem syl6ibr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl6ibr_1 (wi ph (wi ps ch)))
    (syl6ibr_2 (wb th ch)))
  (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.)
(pub theorem syl6bi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl6bi_1 (wi ph (wb ps ch)))
    (syl6bi_2 (wi ch th)))
  (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.)
(pub theorem syl6bir ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl6bir_1 (wi ph (wb ch ps)))
    (syl6bir_2 (wi ch th)))
  (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.)
(pub theorem syl7bi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl7bi_1 (wb ph ps))
    (syl7bi_2 (wi ch (wi th (wi ps ta)))))
  (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.)
(pub theorem syl8ib ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl8ib_1 (wi ph (wi ps (wi ch th))))
    (syl8ib_2 (wb th ta)))
  (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.)
(pub theorem mpbird ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpbird_min (wi ph ch))
    (mpbird_maj (wi ph (wb ps ch))))
  (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.)
(pub theorem mpbiri ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpbiri_min ch)
    (mpbiri_maj (wi ph (wb ps ch))))
  (wi ph ps)
  ()
  (mpbird (ph ps ch)
    (a1i (ch ph)
      mpbiri_min)
    mpbiri_maj))

--| A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(pub theorem sylibrd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylibrd_1 (wi ph (wi ps ch)))
    (sylibrd_2 (wi ph (wb th ch))))
  (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.)
(pub theorem sylbird ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylbird_1 (wi ph (wb ch ps)))
    (sylbird_2 (wi ph (wi ch th))))
  (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.)
(pub theorem biid ((ph wff ()))
  ()
  (wb ph ph)
  ()
  (impbii (ph ph)
    (id (ph))
    (id (ph))))

--| Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)
(pub theorem biidd ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_1im ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem _2th ((ph wff ()) (ps wff ()))
  ((_2th_1 ph)
    (_2th_2 ps))
  (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.)
(pub theorem _2thd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_2thd_1 (wi ph ps))
    (_2thd_2 (wi ph ch)))
  (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.)
(pub theorem ibi ((ph wff ()) (ps wff ()))
  ((ibi_1 (wi ph (wb ph ps))))
  (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.)
(pub theorem ibir ((ph wff ()) (ps wff ()))
  ((ibir_1 (wi ph (wb ps ph))))
  (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.)
(pub theorem ibd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ibd_1 (wi ph (wi ps (wb ps ch)))))
  (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.)
(pub theorem pm5_74 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_74i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm5_74i_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem pm5_74ri ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm5_74ri_1 (wb (wi ph ps) (wi ph ch))))
  (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.)
(pub theorem pm5_74d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((pm5_74d_1 (wi ph (wi ps (wb ch th)))))
  (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.)
(pub theorem pm5_74rd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((pm5_74rd_1 (wi ph (wb (wi ps ch) (wi ps th)))))
  (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.)
(pub theorem bitri ((ph wff ()) (ps wff ()) (ch wff ()))
  ((bitri_1 (wb ph ps))
    (bitri_2 (wb ps ch)))
  (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.)
(pub theorem bitr2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((bitr2i_1 (wb ph ps))
    (bitr2i_2 (wb ps ch)))
  (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.)
(pub theorem bitr3i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((bitr3i_1 (wb ps ph))
    (bitr3i_2 (wb ps ch)))
  (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.)
(pub theorem bitr4i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((bitr4i_1 (wb ph ps))
    (bitr4i_2 (wb ch ps)))
  (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.)
(pub theorem bitrd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((bitrd_1 (wi ph (wb ps ch)))
    (bitrd_2 (wi ph (wb ch th))))
  (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.)
(pub theorem bitr2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((bitr2d_1 (wi ph (wb ps ch)))
    (bitr2d_2 (wi ph (wb ch th))))
  (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.)
(pub theorem bitr3d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((bitr3d_1 (wi ph (wb ps ch)))
    (bitr3d_2 (wi ph (wb ps th))))
  (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.)
(pub theorem bitr4d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((bitr4d_1 (wi ph (wb ps ch)))
    (bitr4d_2 (wi ph (wb th ch))))
  (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.)
(pub theorem syl5bb ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5bb_1 (wb ph ps))
    (syl5bb_2 (wi ch (wb ps th))))
  (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.)
(pub theorem syl5rbb ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5rbb_1 (wb ph ps))
    (syl5rbb_2 (wi ch (wb ps th))))
  (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.)
(pub theorem syl5bbr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5bbr_1 (wb ps ph))
    (syl5bbr_2 (wi ch (wb ps th))))
  (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.)
(pub theorem syl5rbbr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl5rbbr_1 (wb ps ph))
    (syl5rbbr_2 (wi ch (wb ps th))))
  (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.)
(pub theorem syl6bb ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl6bb_1 (wi ph (wb ps ch)))
    (syl6bb_2 (wb ch th)))
  (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.)
(pub theorem syl6rbb ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl6rbb_1 (wi ph (wb ps ch)))
    (syl6rbb_2 (wb ch th)))
  (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.)
(pub theorem syl6bbr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl6bbr_1 (wi ph (wb ps ch)))
    (syl6bbr_2 (wb th ch)))
  (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.)
(pub theorem syl6rbbr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl6rbbr_1 (wi ph (wb ps ch)))
    (syl6rbbr_2 (wb th ch)))
  (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.)
(pub theorem _3imtr3i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3imtr3_1 (wi ph ps))
    (_3imtr3_2 (wb ph ch))
    (_3imtr3_3 (wb ps th)))
  (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.)
(pub theorem _3imtr4i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3imtr4_1 (wi ph ps))
    (_3imtr4_2 (wb ch ph))
    (_3imtr4_3 (wb th ps)))
  (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.)
(pub theorem _3imtr3d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3imtr3d_1 (wi ph (wi ps ch)))
    (_3imtr3d_2 (wi ph (wb ps th)))
    (_3imtr3d_3 (wi ph (wb ch ta))))
  (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.)
(pub theorem _3imtr4d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3imtr4d_1 (wi ph (wi ps ch)))
    (_3imtr4d_2 (wi ph (wb th ps)))
    (_3imtr4d_3 (wi ph (wb ta ch))))
  (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.)
(pub theorem _3imtr3g ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3imtr3g_1 (wi ph (wi ps ch)))
    (_3imtr3g_2 (wb ps th))
    (_3imtr3g_3 (wb ch ta)))
  (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.)
(pub theorem _3imtr4g ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3imtr4g_1 (wi ph (wi ps ch)))
    (_3imtr4g_2 (wb th ps))
    (_3imtr4g_3 (wb ta ch)))
  (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.)
(pub theorem _3bitri ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3bitri_1 (wb ph ps))
    (_3bitri_2 (wb ps ch))
    (_3bitri_3 (wb ch th)))
  (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.)
(pub theorem _3bitrri ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3bitri_1 (wb ph ps))
    (_3bitri_2 (wb ps ch))
    (_3bitri_3 (wb ch th)))
  (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.)
(pub theorem _3bitr2i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3bitr2i_1 (wb ph ps))
    (_3bitr2i_2 (wb ch ps))
    (_3bitr2i_3 (wb ch th)))
  (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.)
(pub theorem _3bitr2ri ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3bitr2i_1 (wb ph ps))
    (_3bitr2i_2 (wb ch ps))
    (_3bitr2i_3 (wb ch th)))
  (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.)
(pub theorem _3bitr3i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3bitr3i_1 (wb ph ps))
    (_3bitr3i_2 (wb ph ch))
    (_3bitr3i_3 (wb ps th)))
  (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.)
(pub theorem _3bitr3ri ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3bitr3i_1 (wb ph ps))
    (_3bitr3i_2 (wb ph ch))
    (_3bitr3i_3 (wb ps th)))
  (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.)
(pub theorem _3bitr4i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3bitr4i_1 (wb ph ps))
    (_3bitr4i_2 (wb ch ph))
    (_3bitr4i_3 (wb th ps)))
  (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.)
(pub theorem _3bitr4ri ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3bitr4i_1 (wb ph ps))
    (_3bitr4i_2 (wb ch ph))
    (_3bitr4i_3 (wb th ps)))
  (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.)
(pub theorem _3bitrd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3bitrd_1 (wi ph (wb ps ch)))
    (_3bitrd_2 (wi ph (wb ch th)))
    (_3bitrd_3 (wi ph (wb th ta))))
  (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.)
(pub theorem _3bitrrd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3bitrd_1 (wi ph (wb ps ch)))
    (_3bitrd_2 (wi ph (wb ch th)))
    (_3bitrd_3 (wi ph (wb th ta))))
  (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.)
(pub theorem _3bitr2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3bitr2d_1 (wi ph (wb ps ch)))
    (_3bitr2d_2 (wi ph (wb th ch)))
    (_3bitr2d_3 (wi ph (wb th ta))))
  (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.)
(pub theorem _3bitr2rd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3bitr2d_1 (wi ph (wb ps ch)))
    (_3bitr2d_2 (wi ph (wb th ch)))
    (_3bitr2d_3 (wi ph (wb th ta))))
  (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.)
(pub theorem _3bitr3d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3bitr3d_1 (wi ph (wb ps ch)))
    (_3bitr3d_2 (wi ph (wb ps th)))
    (_3bitr3d_3 (wi ph (wb ch ta))))
  (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.)
(pub theorem _3bitr3rd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3bitr3d_1 (wi ph (wb ps ch)))
    (_3bitr3d_2 (wi ph (wb ps th)))
    (_3bitr3d_3 (wi ph (wb ch ta))))
  (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.)
(pub theorem _3bitr4d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3bitr4d_1 (wi ph (wb ps ch)))
    (_3bitr4d_2 (wi ph (wb th ps)))
    (_3bitr4d_3 (wi ph (wb ta ch))))
  (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.)
(pub theorem _3bitr4rd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3bitr4d_1 (wi ph (wb ps ch)))
    (_3bitr4d_2 (wi ph (wb th ps)))
    (_3bitr4d_3 (wi ph (wb ta ch))))
  (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.)
(pub theorem _3bitr3g ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3bitr3g_1 (wi ph (wb ps ch)))
    (_3bitr3g_2 (wb ps th))
    (_3bitr3g_3 (wb ch ta)))
  (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.)
(pub theorem _3bitr4g ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3bitr4g_1 (wi ph (wb ps ch)))
    (_3bitr4g_2 (wb th ps))
    (_3bitr4g_3 (wb ta ch)))
  (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.)
(pub theorem notnotb ((ph wff ()))
  ()
  (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.)
(pub theorem con34b ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem con4bid ((ph wff ()) (ps wff ()) (ch wff ()))
  ((con4bid_1 (wi ph (wb (wn ps) (wn ch)))))
  (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.)
(pub theorem notbid ((ph wff ()) (ps wff ()) (ch wff ()))
  ((notbid_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem notbi ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem notbii ((ph wff ()) (ps wff ()))
  ((notbii_1 (wb ph ps)))
  (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.)
(pub theorem con4bii ((ph wff ()) (ps wff ()))
  ((con4bii_1 (wb (wn ph) (wn ps))))
  (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.)
(pub theorem mtbi ((ph wff ()) (ps wff ()))
  ((mtbi_1 (wn ph))
    (mtbi_2 (wb ph ps)))
  (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.)
(pub theorem mtbir ((ph wff ()) (ps wff ()))
  ((mtbir_1 (wn ps))
    (mtbir_2 (wb ph ps)))
  (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.)
(pub theorem mtbid ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mtbid_min (wi ph (wn ps)))
    (mtbid_maj (wi ph (wb ps ch))))
  (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.)
(pub theorem mtbird ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mtbird_min (wi ph (wn ch)))
    (mtbird_maj (wi ph (wb ps ch))))
  (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.)
(pub theorem mtbii ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mtbii_min (wn ps))
    (mtbii_maj (wi ph (wb ps ch))))
  (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.)
(pub theorem mtbiri ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mtbiri_min (wn ch))
    (mtbiri_maj (wi ph (wb ps ch))))
  (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.)
(pub theorem sylnib ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylnib_1 (wi ph (wn ps)))
    (sylnib_2 (wb ps ch)))
  (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.)
(pub theorem sylnibr ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylnibr_1 (wi ph (wn ps)))
    (sylnibr_2 (wb ch ps)))
  (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.)
(pub theorem sylnbi ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylnbi_1 (wb ph ps))
    (sylnbi_2 (wi (wn ps) ch)))
  (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.)
(pub theorem sylnbir ((ph wff ()) (ps wff ()) (ch wff ()))
  ((sylnbir_1 (wb ps ph))
    (sylnbir_2 (wi (wn ps) ch)))
  (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.)
(pub theorem xchnxbi ((ph wff ()) (ps wff ()) (ch wff ()))
  ((xchnxbi_1 (wb (wn ph) ps))
    (xchnxbi_2 (wb ph ch)))
  (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.)
(pub theorem xchnxbir ((ph wff ()) (ps wff ()) (ch wff ()))
  ((xchnxbir_1 (wb (wn ph) ps))
    (xchnxbir_2 (wb ch ph)))
  (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.)
(pub theorem xchbinx ((ph wff ()) (ps wff ()) (ch wff ()))
  ((xchbinx_1 (wb ph (wn ps)))
    (xchbinx_2 (wb ps ch)))
  (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.)
(pub theorem xchbinxr ((ph wff ()) (ps wff ()) (ch wff ()))
  ((xchbinxr_1 (wb ph (wn ps)))
    (xchbinxr_2 (wb ch ps)))
  (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.)
(pub theorem imbi2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((imbi2i_1 (wb ph ps)))
  (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.)
(pub theorem bibi2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((bibi2i_1 (wb ph ps)))
  (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.)
(pub theorem bibi1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((bibi2i_1 (wb ph ps)))
  (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.)
(pub theorem bibi12i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((bibi2i_1 (wb ph ps))
    (bibi12i_2 (wb ch th)))
  (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.)
(pub theorem imbi2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((imbid_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem imbi1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((imbid_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem bibi2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((imbid_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem bibi1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((imbid_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem imbi12d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imbi12d_1 (wi ph (wb ps ch)))
    (imbi12d_2 (wi ph (wb th ta))))
  (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.)
(pub theorem bibi12d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imbi12d_1 (wi ph (wb ps ch)))
    (imbi12d_2 (wi ph (wb th ta))))
  (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.)
(pub theorem imbi12 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem imbi1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem imbi2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem imbi1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((imbi1i_1 (wb ph ps)))
  (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.)
(pub theorem imbi12i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((imbi12i_1 (wb ph ps))
    (imbi12i_2 (wb ch th)))
  (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.)
(pub theorem bibi1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem bitr3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem con2bi ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem con2bid ((ph wff ()) (ps wff ()) (ch wff ()))
  ((con2bid_1 (wi ph (wb ps (wn ch)))))
  (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.)
(pub theorem con1bid ((ph wff ()) (ps wff ()) (ch wff ()))
  ((con1bid_1 (wi ph (wb (wn ps) ch))))
  (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.)
(pub theorem con1bii ((ph wff ()) (ps wff ()))
  ((con1bii_1 (wb (wn ph) ps)))
  (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.)
(pub theorem con2bii ((ph wff ()) (ps wff ()))
  ((con2bii_1 (wb ph (wn ps))))
  (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.)
(pub theorem con1b ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem con2b ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem biimt ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_5 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem a1bi ((ph wff ()) (ps wff ()))
  ((a1bi_1 ph))
  (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.)
(pub theorem mt2bi ((ph wff ()) (ps wff ()))
  ((mt2bi_1 ph))
  (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.)
(pub theorem mtt ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem imnot ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_501 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem ibib ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem ibibr ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem tbt ((ph wff ()) (ps wff ()))
  ((tbt_1 ph))
  (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.)
(pub theorem nbn2 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem bibif ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem nbn ((ph wff ()) (ps wff ()))
  ((nbn_1 (wn ph)))
  (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.)
(pub theorem nbn3 ((ph wff ()) (ps wff ()))
  ((nbn3_1 ph))
  (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.)
(pub theorem pm5_21im ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem _2false ((ph wff ()) (ps wff ()))
  ((_2false_1 (wn ph))
    (_2false_2 (wn ps)))
  (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.)
(pub theorem _2falsed ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_2falsed_1 (wi ph (wn ps)))
    (_2falsed_2 (wi ph (wn ch))))
  (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.)
(pub theorem pm5_21ni ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm5_21ni_1 (wi ph ps))
    (pm5_21ni_2 (wi ch ps)))
  (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.)
(pub theorem pm5_21nii ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm5_21ni_1 (wi ph ps))
    (pm5_21ni_2 (wi ch ps))
    (pm5_21nii_3 (wi ps (wb ph ch))))
  (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.)
(pub theorem pm5_21ndd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((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)))))
  (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.)
(pub theorem bija ((ph wff ()) (ps wff ()) (ch wff ()))
  ((bija_1 (wi ph (wi ps ch)))
    (bija_2 (wi (wn ph) (wi (wn ps) ch))))
  (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.)
(pub theorem pm5_18 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xor3 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem nbbn ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem biass ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_19 ((ph wff ()))
  ()
  (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.)
(pub theorem bi2_04 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_4 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem imdi ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_41 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm4_8 ((ph wff ()))
  ()
  (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.)
(pub theorem pm4_81 ((ph wff ()))
  ()
  (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.)
(pub theorem imim21b ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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 ((ph wff ()) (ps wff ())) (wff ()))

--| Extend wff definition to include conjunction ('and').
(term wa ((ph wff ()) (ps 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.)
(pub theorem pm4_64 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_53 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_54 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem ori ((ph wff ()) (ps wff ()))
  ((ori_1 (wo ph ps)))
  (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.)
(pub theorem orri ((ph wff ()) (ps wff ()))
  ((orri_1 (wi (wn ph) ps)))
  (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.)
(pub theorem ord ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ord_1 (wi ph (wo ps ch))))
  (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.)
(pub theorem orrd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((orrd_1 (wi ph (wi (wn ps) ch))))
  (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.)
(pub theorem jaoi ((ph wff ()) (ps wff ()) (ch wff ()))
  ((jaoi_1 (wi ph ps))
    (jaoi_2 (wi ch ps)))
  (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.)
(pub theorem jaod ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jaod_1 (wi ph (wi ps ch)))
    (jaod_2 (wi ph (wi th ch))))
  (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.)
(pub theorem mpjaod ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jaod_1 (wi ph (wi ps ch)))
    (jaod_2 (wi ph (wi th ch)))
    (jaod_3 (wi ph (wo ps th))))
  (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.)
(pub theorem orel1 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem orel2 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem olc ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem orc ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm1_4 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem orcom ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem orcomd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((orcomd_1 (wi ph (wo ps ch))))
  (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.)
(pub theorem orcoms ((ph wff ()) (ps wff ()) (ch wff ()))
  ((orcoms_1 (wi (wo ph ps) ch)))
  (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.)
(pub theorem orci ((ph wff ()) (ps wff ()))
  ((orci_1 ph))
  (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.)
(pub theorem olci ((ph wff ()) (ps wff ()))
  ((orci_1 ph))
  (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.)
(pub theorem orcd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((orcd_1 (wi ph ps)))
  (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.)
(pub theorem olcd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((orcd_1 (wi ph ps)))
  (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.)
(pub theorem orcs ((ph wff ()) (ps wff ()) (ch wff ()))
  ((orcs_1 (wi (wo ph ps) ch)))
  (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.)
(pub theorem olcs ((ph wff ()) (ps wff ()) (ch wff ()))
  ((olcs_1 (wi (wo ph ps) ch)))
  (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.)
(pub theorem pm2_07 ((ph wff ()))
  ()
  (wi ph (wo ph ph))
  ()
  (olc (ph ph)))

--| Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(pub theorem pm2_45 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_46 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_47 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_48 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_49 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_67_2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_67 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_25 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem biorf ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem biortn ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem biorfi ((ph wff ()) (ps wff ()))
  ((biorfi_1 (wn ph)))
  (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.)
(pub theorem biorfiOLD ((ph wff ()) (ps wff ()))
  ((biorfi_1 (wn ph)))
  (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.)
(pub theorem pm2_621 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_62 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_68 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem dfor2 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem imor ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem imori ((ph wff ()) (ps wff ()))
  ((imori_1 (wi ph ps)))
  (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.)
(pub theorem imorri ((ph wff ()) (ps wff ()))
  ((imorri_1 (wo (wn ph) ps)))
  (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.)
(pub theorem exmid ((ph wff ()))
  ()
  (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.)
(pub theorem exmidd ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_1 ((ph wff ()))
  ()
  (wo (wn ph) ph)
  ()
  (imori (ph ph)
    (id (ph))))

--| Theorem *2.13 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
(pub theorem pm2_13 ((ph wff ()))
  ()
  (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.)
(pub theorem pm4_62 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_66 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_63 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem imnan ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem imnani ((ph wff ()) (ps wff ()))
  ((imnani_1 (wn (wa ph ps))))
  (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.)
(pub theorem iman ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem annim ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_61 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_65 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_67 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem imp ((ph wff ()) (ps wff ()) (ch wff ()))
  ((imp_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem impcom ((ph wff ()) (ps wff ()) (ch wff ()))
  ((imp_1 (wi ph (wi ps ch))))
  (wi (wa ps ph) ch)
  ()
  (imp (ps ph ch)
    (com12 (ph ps ch)
      imp_1)))

--| Importation deduction. (Contributed by NM, 31-Mar-1994.)
(pub theorem impd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((impd_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem imp31 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((impd_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem imp32 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((impd_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem ex ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ex_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem expcom ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ex_1 (wi (wa ph ps) ch)))
  (wi ps (wi ph ch))
  ()
  (com12 (ph ps ch)
    (ex (ph ps ch)
      ex_1)))

--| Exportation deduction. (Contributed by NM, 20-Aug-1993.)
(pub theorem expd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((expd_1 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem expdimp ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((expd_1 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem expcomd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((expcomd_1 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem expdcom ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((expdcom_1 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem impancom ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((impancom_1 (wi (wa ph ps) (wi ch th))))
  (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.)
(pub theorem con3dimp ((ph wff ()) (ps wff ()) (ch wff ()))
  ((con3dimp_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem pm2_01da ((ph wff ()) (ps wff ()))
  ((pm2_01da_1 (wi (wa ph ps) (wn ps))))
  (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.)
(pub theorem pm2_18da ((ph wff ()) (ps wff ()))
  ((pm2_18da_1 (wi (wa ph (wn ps)) ps)))
  (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.)
(pub theorem pm3_3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm3_31 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem impexp ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm3_2 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm3_21 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm3_22 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem ancom ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem ancomd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ancomd_1 (wi ph (wa ps ch))))
  (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.)
(pub theorem ancomst ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem ancoms ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ancoms_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ancomsd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((ancomsd_1 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem pm3_2i ((ph wff ()) (ps wff ()))
  ((pm3_2i_1 ph)
    (pm3_2i_2 ps))
  (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.)
(pub theorem pm3_43i ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem adantr ((ph wff ()) (ps wff ()) (ch wff ()))
  ((adantr_1 (wi ph ps)))
  (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.)
(pub theorem adantl ((ph wff ()) (ps wff ()) (ch wff ()))
  ((adantl_1 (wi ph ps)))
  (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.)
(pub theorem simpl ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem simplOLD ((ph wff ()) (ps wff ()))
  ()
  (wi (wa ph ps) ph)
  ()
  (imp (ph ps ph)
    (ax_1 (ph ps))))

--| Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
(pub theorem simpli ((ph wff ()) (ps wff ()))
  ((simpli_1 (wa ph ps)))
  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.)
(pub theorem simpld ((ph wff ()) (ps wff ()) (ch wff ()))
  ((simpld_1 (wi ph (wa ps ch))))
  (wi ph ps)
  ()
  (syl (ph (wa ps ch) ps)
    simpld_1
    (simpl (ps ch))))

--| Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(pub theorem simplbi ((ph wff ()) (ps wff ()) (ch wff ()))
  ((simplbi_1 (wb ph (wa ps ch))))
  (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.)
(pub theorem simpr ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem simprOLD ((ph wff ()) (ps wff ()))
  ()
  (wi (wa ph ps) ps)
  ()
  (imp (ph ps ps)
    (idd (ph ps))))

--| Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
(pub theorem simpri ((ph wff ()) (ps wff ()))
  ((simpri_1 (wa ph ps)))
  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.)
(pub theorem simprd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((simprd_1 (wi ph (wa ps ch))))
  (wi ph ch)
  ()
  (simpld (ph ch ps)
    (ancomd (ph ps ch)
      simprd_1)))

--| Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(pub theorem simprbi ((ph wff ()) (ps wff ()) (ch wff ()))
  ((simprbi_1 (wb ph (wa ps ch))))
  (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.)
(pub theorem adantld ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((adantld_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem adantrd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((adantrd_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem impel ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((impel_1 (wi ph (wi ps ch)))
    (impel_2 (wi th ps)))
  (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.)
(pub theorem mpan9 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpan9_1 (wi ph ps))
    (mpan9_2 (wi ch (wi ps th))))
  (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.)
(pub theorem syldan ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syldan_1 (wi (wa ph ps) ch))
    (syldan_2 (wi (wa ph ch) th)))
  (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.)
(pub theorem sylan ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylan_1 (wi ph ps))
    (sylan_2 (wi (wa ps ch) th)))
  (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.)
(pub theorem sylanb ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylanb_1 (wb ph ps))
    (sylanb_2 (wi (wa ps ch) th)))
  (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.)
(pub theorem sylanbr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylanbr_1 (wb ps ph))
    (sylanbr_2 (wi (wa ps ch) th)))
  (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.)
(pub theorem sylan2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylan2_1 (wi ph ch))
    (sylan2_2 (wi (wa ps ch) th)))
  (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.)
(pub theorem sylan2b ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylan2b_1 (wb ph ch))
    (sylan2b_2 (wi (wa ps ch) th)))
  (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.)
(pub theorem sylan2br ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylan2br_1 (wb ch ph))
    (sylan2br_2 (wi (wa ps ch) th)))
  (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.)
(pub theorem syl2an ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl2an_1 (wi ph ps))
    (syl2an_2 (wi ta ch))
    (syl2an_3 (wi (wa ps ch) th)))
  (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.)
(pub theorem syl2anr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl2an_1 (wi ph ps))
    (syl2an_2 (wi ta ch))
    (syl2an_3 (wi (wa ps ch) th)))
  (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.)
(pub theorem syl2anb ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl2anb_1 (wb ph ps))
    (syl2anb_2 (wb ta ch))
    (syl2anb_3 (wi (wa ps ch) th)))
  (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.)
(pub theorem syl2anbr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl2anbr_1 (wb ps ph))
    (syl2anbr_2 (wb ch ta))
    (syl2anbr_3 (wi (wa ps ch) th)))
  (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.)
(pub theorem syland ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syland_1 (wi ph (wi ps ch)))
    (syland_2 (wi ph (wi (wa ch th) ta))))
  (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.)
(pub theorem sylan2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylan2d_1 (wi ph (wi ps ch)))
    (sylan2d_2 (wi ph (wi (wa th ch) ta))))
  (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.)
(pub theorem syl2and ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((syl2and_1 (wi ph (wi ps ch)))
    (syl2and_2 (wi ph (wi th ta)))
    (syl2and_3 (wi ph (wi (wa ch ta) et))))
  (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.)
(pub theorem biimpa ((ph wff ()) (ps wff ()) (ch wff ()))
  ((biimpa_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem biimpar ((ph wff ()) (ps wff ()) (ch wff ()))
  ((biimpa_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem biimpac ((ph wff ()) (ps wff ()) (ch wff ()))
  ((biimpa_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem biimparc ((ph wff ()) (ps wff ()) (ch wff ()))
  ((biimpa_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem animorl ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem animorr ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem animorlr ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem animorrl ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem ianor ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem anor ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem ioran ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_52 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_53 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_54 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_55 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_56 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem oran ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_57 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm3_1 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm3_11 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm3_12 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm3_13 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm3_14 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem iba ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem ibar ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem biantru ((ph wff ()) (ps wff ()))
  ((biantru_1 ph))
  (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.)
(pub theorem biantrur ((ph wff ()) (ps wff ()))
  ((biantrur_1 ph))
  (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.)
(pub theorem biantrud ((ph wff ()) (ps wff ()) (ch wff ()))
  ((biantrud_1 (wi ph ps)))
  (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.)
(pub theorem biantrurd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((biantrud_1 (wi ph ps)))
  (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.)
(pub theorem mpbirand ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpbirand_1 (wi ph ch))
    (mpbirand_2 (wi ph (wb ps (wa ch th)))))
  (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.)
(pub theorem jaao ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((jaao_1 (wi ph (wi ps ch)))
    (jaao_2 (wi th (wi ta ch))))
  (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.)
(pub theorem jaoa ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((jaao_1 (wi ph (wi ps ch)))
    (jaao_2 (wi th (wi ta ch))))
  (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.)
(pub theorem pm3_44 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem jao ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm1_2 ((ph wff ()))
  ()
  (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.)
(pub theorem oridm ((ph wff ()))
  ()
  (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.)
(pub theorem pm4_25 ((ph wff ()))
  ()
  (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.)
(pub theorem orim12i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((orim12i_1 (wi ph ps))
    (orim12i_2 (wi ch th)))
  (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.)
(pub theorem orim1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((orim1i_1 (wi ph ps)))
  (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.)
(pub theorem orim2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((orim1i_1 (wi ph ps)))
  (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.)
(pub theorem orbi2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((orbi2i_1 (wb ph ps)))
  (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.)
(pub theorem orbi1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((orbi2i_1 (wb ph ps)))
  (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.)
(pub theorem orbi12i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((orbi12i_1 (wb ph ps))
    (orbi12i_2 (wb ch th)))
  (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.)
(pub theorem pm1_5 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem or12 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem orass ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_31 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_32 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem or32 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem or4 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem or42 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem orordi ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem orordir ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem jca ((ph wff ()) (ps wff ()) (ch wff ()))
  ((jca_1 (wi ph ps))
    (jca_2 (wi ph ch)))
  (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.)
(pub theorem jcad ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jcad_1 (wi ph (wi ps ch)))
    (jcad_2 (wi ph (wi ps th))))
  (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.)
(pub theorem jca2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jca2_1 (wi ph (wi ps ch)))
    (jca2_2 (wi ps th)))
  (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.)
(pub theorem jca31 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jca31_1 (wi ph ps))
    (jca31_2 (wi ph ch))
    (jca31_3 (wi ph th)))
  (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.)
(pub theorem jca32 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jca31_1 (wi ph ps))
    (jca31_2 (wi ph ch))
    (jca31_3 (wi ph th)))
  (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.)
(pub theorem jcai ((ph wff ()) (ps wff ()) (ch wff ()))
  ((jcai_1 (wi ph ps))
    (jcai_2 (wi ph (wi ps ch))))
  (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.)
(pub theorem jctil ((ph wff ()) (ps wff ()) (ch wff ()))
  ((jctil_1 (wi ph ps))
    (jctil_2 ch))
  (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.)
(pub theorem jctir ((ph wff ()) (ps wff ()) (ch wff ()))
  ((jctil_1 (wi ph ps))
    (jctil_2 ch))
  (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.)
(pub theorem jccir ((ph wff ()) (ps wff ()) (ch wff ()))
  ((jccir_1 (wi ph ps))
    (jccir_2 (wi ps ch)))
  (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.)
(pub theorem jccil ((ph wff ()) (ps wff ()) (ch wff ()))
  ((jccir_1 (wi ph ps))
    (jccir_2 (wi ps ch)))
  (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.)
(pub theorem jctl ((ph wff ()) (ps wff ()))
  ((jctl_1 ps))
  (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.)
(pub theorem jctr ((ph wff ()) (ps wff ()))
  ((jctl_1 ps))
  (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.)
(pub theorem jctild ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jctild_1 (wi ph (wi ps ch)))
    (jctild_2 (wi ph th)))
  (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.)
(pub theorem jctird ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jctird_1 (wi ph (wi ps ch)))
    (jctird_2 (wi ph th)))
  (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.)
(pub theorem syl6an ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl6an_1 (wi ph ps))
    (syl6an_2 (wi ph (wi ch th)))
    (syl6an_3 (wi (wa ps th) ta)))
  (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.)
(pub theorem ancl ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem anclb ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_42 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem ancr ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem ancrb ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem ancli ((ph wff ()) (ps wff ()))
  ((ancli_1 (wi ph ps)))
  (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.)
(pub theorem ancri ((ph wff ()) (ps wff ()))
  ((ancri_1 (wi ph ps)))
  (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.)
(pub theorem ancld ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ancld_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem ancrd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ancrd_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem anc2l ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem anc2r ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem anc2li ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anc2li_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem anc2ri ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anc2ri_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem pm3_41 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm3_42 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm3_4 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_45im ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem anim12d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((anim12d_1 (wi ph (wi ps ch)))
    (anim12d_2 (wi ph (wi th ta))))
  (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.)
(pub theorem anim12d1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((anim12d1_1 (wi ph (wi ps ch)))
    (anim12d1_2 (wi th ta)))
  (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.)
(pub theorem anim1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((anim1d_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem anim2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((anim1d_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem anim12i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((anim12i_1 (wi ph ps))
    (anim12i_2 (wi ch th)))
  (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.)
(pub theorem anim12ci ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((anim12i_1 (wi ph ps))
    (anim12i_2 (wi ch th)))
  (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.)
(pub theorem anim1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anim1i_1 (wi ph ps)))
  (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.)
(pub theorem anim2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anim1i_1 (wi ph ps)))
  (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.)
(pub theorem anim12ii ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((anim12ii_1 (wi ph (wi ps ch)))
    (anim12ii_2 (wi th (wi ps ta))))
  (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.)
(pub theorem prth ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem pm2_3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_41 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_42 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_4 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_65da ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_65da_1 (wi (wa ph ps) ch))
    (pm2_65da_2 (wi (wa ph ps) (wn ch))))
  (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.)
(pub theorem pm4_44 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_14 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm3_37 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem nan ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm4_15 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm4_78 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm4_79 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm4_87 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm3_33 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm3_34 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm3_35 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_31 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem imp4b ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imp4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem imp4a ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imp4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem imp4aOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imp4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem imp4bOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imp4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem imp4c ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imp4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem imp4d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imp4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem imp41 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imp4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem imp42 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imp4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem imp43 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imp4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem imp44 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imp4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem imp45 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((imp4_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem imp5a ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((imp5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem imp5d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((imp5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem imp5g ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((imp5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem imp55 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((imp5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem imp511 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((imp5_1 (wi ph (wi ps (wi ch (wi th (wi ta et)))))))
  (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.)
(pub theorem expimpd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((expimpd_1 (wi (wa ph ps) (wi ch th))))
  (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.)
(pub theorem exp31 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((exp31_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem exp32 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((exp32_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem exp4b ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((exp4b_1 (wi (wa ph ps) (wi (wa ch th) ta))))
  (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.)
(pub theorem exp4a ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((exp4a_1 (wi ph (wi ps (wi (wa ch th) ta)))))
  (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.)
(pub theorem exp4aOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((exp4aOLD_1 (wi ph (wi ps (wi (wa ch th) ta)))))
  (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.)
(pub theorem exp4bOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((exp4bOLD_1 (wi (wa ph ps) (wi (wa ch th) ta))))
  (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.)
(pub theorem exp4c ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((exp4c_1 (wi ph (wi (wa (wa ps ch) th) ta))))
  (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.)
(pub theorem exp4d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((exp4d_1 (wi ph (wi (wa ps (wa ch th)) ta))))
  (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.)
(pub theorem exp41 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((exp41_1 (wi (wa (wa (wa ph ps) ch) th) ta)))
  (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.)
(pub theorem exp42 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((exp42_1 (wi (wa (wa ph (wa ps ch)) th) ta)))
  (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.)
(pub theorem exp43 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((exp43_1 (wi (wa (wa ph ps) (wa ch th)) ta)))
  (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.)
(pub theorem exp44 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((exp44_1 (wi (wa ph (wa (wa ps ch) th)) ta)))
  (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.)
(pub theorem exp45 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((exp45_1 (wi (wa ph (wa ps (wa ch th))) ta)))
  (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.)
(pub theorem expr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((expr_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem exp5c ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((exp5c_1 (wi ph (wi (wa ps ch) (wi (wa th ta) et)))))
  (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.)
(pub theorem exp5j ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((exp5j_1 (wi ph (wi (wa (wa (wa ps ch) th) ta) et))))
  (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.)
(pub theorem exp5l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((exp5l_1 (wi ph (wi (wa (wa ps ch) (wa th ta)) et))))
  (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.)
(pub theorem exp53 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((exp53_1 (wi (wa (wa (wa ph ps) (wa ch th)) ta) et)))
  (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.)
(pub theorem expl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((expl_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem impr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((impr_1 (wi (wa ph ps) (wi ch th))))
  (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.)
(pub theorem impl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((impl_1 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem impac ((ph wff ()) (ps wff ()) (ch wff ()))
  ((impac_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem exbiri ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((exbiri_1 (wi (wa ph ps) (wb ch th))))
  (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.)
(pub theorem simprbda ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((pm3_26bda_1 (wi ph (wb ps (wa ch th)))))
  (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.)
(pub theorem simplbda ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((pm3_26bda_1 (wi ph (wb ps (wa ch th)))))
  (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.)
(pub theorem simplbi2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((simplbi2_1 (wb ph (wa ps ch))))
  (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.)
(pub theorem simplbi2comt ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem simplbi2com ((ph wff ()) (ps wff ()) (ch wff ()))
  ((simplbi2com_1 (wb ph (wa ps ch))))
  (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.)
(pub theorem simpl2im ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((simpl2im_1 (wi ph (wa ps ch)))
    (simpl2im_2 (wi ch th)))
  (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.)
(pub theorem simpl2imOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((simpl2im_1 (wi ph (wa ps ch)))
    (simpl2im_2 (wi ch th)))
  (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.)
(pub theorem simplbiim ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((simplbiim_1 (wb ph (wa ps ch)))
    (simplbiim_2 (wi ch th)))
  (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.)
(pub theorem simplbiimOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((simplbiim_1 (wb ph (wa ps ch)))
    (simplbiim_2 (wi ch th)))
  (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.)
(pub theorem dfbi2 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem dfbi ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_71 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_71r ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_71i ((ph wff ()) (ps wff ()))
  ((pm4_71i_1 (wi ph ps)))
  (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.)
(pub theorem pm4_71ri ((ph wff ()) (ps wff ()))
  ((pm4_71ri_1 (wi ph ps)))
  (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.)
(pub theorem pm4_71d ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm4_71rd_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem pm4_71rd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm4_71rd_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem pm5_32 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_32i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm5_32i_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem pm5_32ri ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm5_32i_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem pm5_32d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((pm5_32d_1 (wi ph (wi ps (wb ch th)))))
  (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.)
(pub theorem pm5_32rd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((pm5_32d_1 (wi ph (wi ps (wb ch th)))))
  (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.)
(pub theorem pm5_32da ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((pm5_32da_1 (wi (wa ph ps) (wb ch th))))
  (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.)
(pub theorem biadan2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((biadan2_1 (wi ph ps))
    (biadan2_2 (wi ps (wb ph ch))))
  (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.)
(pub theorem pm4_24 ((ph wff ()))
  ()
  (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.)
(pub theorem anidm ((ph wff ()))
  ()
  (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.)
(pub theorem anidms ((ph wff ()) (ps wff ()))
  ((anidms_1 (wi (wa ph ph) ps)))
  (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.)
(pub theorem anidmdbi ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem anasss ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((anasss_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem anassrs ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((anassrs_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem anass ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem sylanl1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylanl1_1 (wi ph ps))
    (sylanl1_2 (wi (wa (wa ps ch) th) ta)))
  (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.)
(pub theorem sylanl2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylanl2_1 (wi ph ch))
    (sylanl2_2 (wi (wa (wa ps ch) th) ta)))
  (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.)
(pub theorem sylanr1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylanr1_1 (wi ph ch))
    (sylanr1_2 (wi (wa ps (wa ch th)) ta)))
  (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.)
(pub theorem sylanr2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylanr2_1 (wi ph th))
    (sylanr2_2 (wi (wa ps (wa ch th)) ta)))
  (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.)
(pub theorem sylani ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylani_1 (wi ph ch))
    (sylani_2 (wi ps (wi (wa ch th) ta))))
  (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.)
(pub theorem sylan2i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylan2i_1 (wi ph th))
    (sylan2i_2 (wi ps (wi (wa ch th) ta))))
  (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.)
(pub theorem syl2ani ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((syl2ani_1 (wi ph ch))
    (syl2ani_2 (wi et th))
    (syl2ani_3 (wi ps (wi (wa ch th) ta))))
  (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.)
(pub theorem sylan9 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylan9_1 (wi ph (wi ps ch)))
    (sylan9_2 (wi th (wi ch ta))))
  (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.)
(pub theorem sylan9r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylan9r_1 (wi ph (wi ps ch)))
    (sylan9r_2 (wi th (wi ch ta))))
  (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.)
(pub theorem mtand ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mtand_1 (wi ph (wn ch)))
    (mtand_2 (wi (wa ph ps) ch)))
  (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.)
(pub theorem mtord ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mtord_1 (wi ph (wn ch)))
    (mtord_2 (wi ph (wn th)))
    (mtord_3 (wi ph (wi ps (wo ch th)))))
  (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.)
(pub theorem syl2anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((syl2anc_1 (wi ph ps))
    (syl2anc_2 (wi ph ch))
    (syl2anc_3 (wi (wa ps ch) th)))
  (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.)
(pub theorem sylancl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylancl_1 (wi ph ps))
    (sylancl_2 ch)
    (sylancl_3 (wi (wa ps ch) th)))
  (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.)
(pub theorem sylancr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylancr_1 ps)
    (sylancr_2 (wi ph ch))
    (sylancr_3 (wi (wa ps ch) th)))
  (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.)
(pub theorem sylanblc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylanblc_1 (wi ph ps))
    (sylanblc_2 ch)
    (sylanblc_3 (wb (wa ps ch) th)))
  (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.)
(pub theorem sylanblrc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylanblrc_1 (wi ph ps))
    (sylanblrc_2 ch)
    (sylanblrc_3 (wb th (wa ps ch))))
  (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.)
(pub theorem sylanbrc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylanbrc_1 (wi ph ps))
    (sylanbrc_2 (wi ph ch))
    (sylanbrc_3 (wb th (wa ps ch))))
  (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.)
(pub theorem sylancb ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylancb_1 (wb ph ps))
    (sylancb_2 (wb ph ch))
    (sylancb_3 (wi (wa ps ch) th)))
  (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.)
(pub theorem sylancbr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylancbr_1 (wb ps ph))
    (sylancbr_2 (wb ch ph))
    (sylancbr_3 (wi (wa ps ch) th)))
  (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.)
(pub theorem sylancom ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((sylancom_1 (wi (wa ph ps) ch))
    (sylancom_2 (wi (wa ch ps) th)))
  (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.)
(pub theorem mpdan ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpdan_1 (wi ph ps))
    (mpdan_2 (wi (wa ph ps) ch)))
  (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.)
(pub theorem mpancom ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpancom_1 (wi ps ph))
    (mpancom_2 (wi (wa ph ps) ch)))
  (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.)
(pub theorem mpidan ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpidan_1 (wi ph ch))
    (mpidan_2 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem mpan ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpan_1 ph)
    (mpan_2 (wi (wa ph ps) ch)))
  (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.)
(pub theorem mpan2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpan2_1 ps)
    (mpan2_2 (wi (wa ph ps) ch)))
  (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.)
(pub theorem mp2an ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mp2an_1 ph)
    (mp2an_2 ps)
    (mp2an_3 (wi (wa ph ps) ch)))
  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.)
(pub theorem mp4an ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mp4an_1 ph)
    (mp4an_2 ps)
    (mp4an_3 ch)
    (mp4an_4 th)
    (mp4an_5 (wi (wa (wa ph ps) (wa ch th)) ta)))
  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.)
(pub theorem mpan2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpan2d_1 (wi ph ch))
    (mpan2d_2 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem mpand ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpand_1 (wi ph ps))
    (mpand_2 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem mpani ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpani_1 ps)
    (mpani_2 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem mpan2i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpan2i_1 ch)
    (mpan2i_2 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem mp2ani ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mp2ani_1 ps)
    (mp2ani_2 ch)
    (mp2ani_3 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem mp2and ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mp2and_1 (wi ph ps))
    (mp2and_2 (wi ph ch))
    (mp2and_3 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem mpanl1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpanl1_1 ph)
    (mpanl1_2 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem mpanl2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpanl2_1 ps)
    (mpanl2_2 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem mpanl12 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpanl12_1 ph)
    (mpanl12_2 ps)
    (mpanl12_3 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem mpanr1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpanr1_1 ps)
    (mpanr1_2 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem mpanr2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpanr2_1 ch)
    (mpanr2_2 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem mpanr12 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpanr12_1 ps)
    (mpanr12_2 ch)
    (mpanr12_3 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem mpanlr1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mpanlr1_1 ps)
    (mpanlr1_2 (wi (wa (wa ph (wa ps ch)) th) ta)))
  (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.)
(pub theorem pm5_74da ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((pm5_74da_1 (wi (wa ph ps) (wb ch th))))
  (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.)
(pub theorem pm4_45 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem imdistan ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem imdistani ((ph wff ()) (ps wff ()) (ch wff ()))
  ((imdistani_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem imdistanri ((ph wff ()) (ps wff ()) (ch wff ()))
  ((imdistanri_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem imdistand ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((imdistand_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem imdistanda ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((imdistanda_1 (wi (wa ph ps) (wi ch th))))
  (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.)
(pub theorem anbi2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anbi_1 (wb ph ps)))
  (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.)
(pub theorem anbi1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anbi_1 (wb ph ps)))
  (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.)
(pub theorem anbi2ci ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anbi_1 (wb ph ps)))
  (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.)
(pub theorem anbi12i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((anbi12_1 (wb ph ps))
    (anbi12_2 (wb ch th)))
  (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.)
(pub theorem anbi12ci ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((anbi12_1 (wb ph ps))
    (anbi12_2 (wb ch th)))
  (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.)
(pub theorem syldanl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syldanl_1 (wi (wa ph ps) ch))
    (syldanl_2 (wi (wa (wa ph ch) th) ta)))
  (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.)
(pub theorem sylan9bb ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylan9bb_1 (wi ph (wb ps ch)))
    (sylan9bb_2 (wi th (wb ch ta))))
  (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.)
(pub theorem sylan9bbr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((sylan9bbr_1 (wi ph (wb ps ch)))
    (sylan9bbr_2 (wi th (wb ch ta))))
  (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.)
(pub theorem orbi2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((bid_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem orbi1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((bid_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem anbi2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((bid_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem anbi1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((bid_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem orbi1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem anbi1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem anbi2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem bitr ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem orbi12d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((bi12d_1 (wi ph (wb ps ch)))
    (bi12d_2 (wi ph (wb th ta))))
  (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.)
(pub theorem anbi12d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((bi12d_1 (wi ph (wb ps ch)))
    (bi12d_2 (wi ph (wb th ta))))
  (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.)
(pub theorem pm5_3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_61 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem adantll ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((adant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem adantlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((adant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem adantrl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((adant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem adantrr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((adant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem adantlll ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((adantl2_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem adantllr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((adantl2_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem adantlrl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((adantl2_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem adantlrr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((adantl2_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem adantrll ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((adantr2_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem adantrlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((adantr2_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem adantrrl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((adantr2_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem adantrrr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((adantr2_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem ad2antrr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad2antlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad2antrl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad2antll ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad3antrrr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad3antlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad3antlrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad4antr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad4antrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad4antlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad4antlrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad5antr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad5antrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad5antlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad5antlrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad6antr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad6antrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad6antlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad6antlrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad7antr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad7antrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad7antlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad7antlrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad8antr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad8antrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad8antlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad8antlrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad9antr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad9antrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad9antlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad9antlrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad10antr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()) (ka wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad10antrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()) (ka wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad10antlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()) (ka wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad10antlrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()) (ka wff ()))
  ((ad2ant_1 (wi ph ps)))
  (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.)
(pub theorem ad2ant2l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad2ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad2ant2r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad2ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad2ant2lr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad2ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad2ant2rl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad2ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem adantl3r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((adantl3r_1 (wi (wa (wa (wa ph ps) ch) th) ta)))
  (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.)
(pub theorem adantl4r ((ph wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()) (ka wff ()))
  ((adantl4r_1 (wi (wa (wa (wa (wa ph si) rh) mu) la) ka)))
  (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.)
(pub theorem adantl5r ((ph wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()) (ka wff ()))
  ((adantl5r_1 (wi (wa (wa (wa (wa (wa ph ze) si) rh) mu) la) ka)))
  (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.)
(pub theorem adantl6r ((ph wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()) (ka wff ()))
  ((adantl6r_1 (wi (wa (wa (wa (wa (wa (wa ph et) ze) si) rh) mu) la) ka)))
  (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.)
(pub theorem simpll ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem simplld ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((simplld_1 (wi ph (wa (wa ps ch) th))))
  (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.)
(pub theorem simplr ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem simplrd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((simplrd_1 (wi ph (wa (wa ps ch) th))))
  (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.)
(pub theorem simprl ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem simprld ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((simprld_1 (wi ph (wa ps (wa ch th)))))
  (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.)
(pub theorem simprr ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem simprrd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((simprrd_1 (wi ph (wa ps (wa ch th)))))
  (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.)
(pub theorem simplll ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simplllOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpllr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpllrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simplrl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simplrr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simprll ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simprlr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simprrl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simprrr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simp_4l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp_4lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp_4r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp_4rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp_5l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp_5lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp_5r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp_5rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp_6l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp_6lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp_6r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp_6rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp_7l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ()
  (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.)
(pub theorem simp_7lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ()
  (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.)
(pub theorem simp_7r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ()
  (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.)
(pub theorem simp_7rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ()
  (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.)
(pub theorem simp_8l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ()
  (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.)
(pub theorem simp_8lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ()
  (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.)
(pub theorem simp_8r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ()
  (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.)
(pub theorem simp_8rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ()
  (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.)
(pub theorem simp_9l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()))
  ()
  (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.)
(pub theorem simp_9lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()))
  ()
  (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.)
(pub theorem simp_9r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()))
  ()
  (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.)
(pub theorem simp_9rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()))
  ()
  (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.)
(pub theorem simp_10l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()))
  ()
  (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.)
(pub theorem simp_10lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()))
  ()
  (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.)
(pub theorem simp_10r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()))
  ()
  (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.)
(pub theorem simp_10rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()))
  ()
  (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.)
(pub theorem simp_11l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()) (ka wff ()))
  ()
  (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.)
(pub theorem simp_11lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()) (ka wff ()))
  ()
  (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.)
(pub theorem simp_11r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()) (ka wff ()))
  ()
  (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.)
(pub theorem simp_11rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()) (ka wff ()))
  ()
  (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.)
(pub theorem jaob ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem adant423OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((adant423_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem jaoian ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jaoian_1 (wi (wa ph ps) ch))
    (jaoian_2 (wi (wa th ps) ch)))
  (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.)
(pub theorem jao1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((jao1i_1 (wi ps (wi ch ph))))
  (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.)
(pub theorem jaodan ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jaodan_1 (wi (wa ph ps) ch))
    (jaodan_2 (wi (wa ph th) ch)))
  (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.)
(pub theorem mpjaodan ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((jaodan_1 (wi (wa ph ps) ch))
    (jaodan_2 (wi (wa ph th) ch))
    (jaodan_3 (wi ph (wo ps th))))
  (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.)
(pub theorem pm4_77 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_63 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_64 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm2_61ian ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_61ian_1 (wi (wa ph ps) ch))
    (pm2_61ian_2 (wi (wa (wn ph) ps) ch)))
  (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.)
(pub theorem pm2_61dan ((ph wff ()) (ps wff ()) (ch wff ()))
  ((pm2_61dan_1 (wi (wa ph ps) ch))
    (pm2_61dan_2 (wi (wa ph (wn ps)) ch)))
  (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.)
(pub theorem pm2_61ddan ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((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)))
  (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.)
(pub theorem pm2_61dda ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((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)))
  (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.)
(pub theorem condan ((ph wff ()) (ps wff ()) (ch wff ()))
  ((condan_1 (wi (wa ph (wn ps)) ch))
    (condan_2 (wi (wa ph (wn ps)) (wn ch))))
  (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.)
(pub theorem abai ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_53 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem an12 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem an32 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem an13 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem an31 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem bianass ((ph wff ()) (ps wff ()) (ch wff ()) (et wff ()))
  ((bianass_1 (wb ph (wa ps ch))))
  (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.)
(pub theorem an12s ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((an12s_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem ancom2s ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((an12s_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem an13s ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((an12s_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem an32s ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((an32s_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem ancom1s ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((an32s_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem an31s ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((an32s_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem anass1rs ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((anass1rs_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem anabs1 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem anabs5 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem anabs7 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem a2and ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (rh wff ()))
  ((a2and_1 (wi ph (wi (wa ps rh) (wi ta th))))
    (a2and_2 (wi ph (wi (wa ps rh) ch))))
  (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.)
(pub theorem anabsan ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anabsan_1 (wi (wa (wa ph ph) ps) ch)))
  (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.)
(pub theorem anabss1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anabss1_1 (wi (wa (wa ph ps) ph) ch)))
  (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.)
(pub theorem anabss4 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anabss4_1 (wi (wa (wa ps ph) ps) ch)))
  (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.)
(pub theorem anabss5 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anabss5_1 (wi (wa ph (wa ph ps)) ch)))
  (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.)
(pub theorem anabsi5 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anabsi5_1 (wi ph (wi (wa ph ps) ch))))
  (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.)
(pub theorem anabsi6 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anabsi6_1 (wi ph (wi (wa ps ph) ch))))
  (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.)
(pub theorem anabsi7 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anabsi7_1 (wi ps (wi (wa ph ps) ch))))
  (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.)
(pub theorem anabsi8 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anabsi8_1 (wi ps (wi (wa ps ph) ch))))
  (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.)
(pub theorem anabss7 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anabss7_1 (wi (wa ps (wa ph ps)) ch)))
  (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.)
(pub theorem anabsan2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anabsan2_1 (wi (wa ph (wa ps ps)) ch)))
  (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.)
(pub theorem anabss3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((anabss3_1 (wi (wa (wa ph ps) ps) ch)))
  (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.)
(pub theorem an4 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem an42 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem an43 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem an3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem an4s ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((an4s_1 (wi (wa (wa ph ps) (wa ch th)) ta)))
  (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.)
(pub theorem an42s ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((an41r3s_1 (wi (wa (wa ph ps) (wa ch th)) ta)))
  (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.)
(pub theorem anandi ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem anandir ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem anandis ((ph wff ()) (ps wff ()) (ch wff ()) (ta wff ()))
  ((anandis_1 (wi (wa (wa ph ps) (wa ph ch)) ta)))
  (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.)
(pub theorem anandirs ((ph wff ()) (ps wff ()) (ch wff ()) (ta wff ()))
  ((anandirs_1 (wi (wa (wa ph ch) (wa ps ch)) ta)))
  (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.)
(pub theorem syl2an2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl2an2_1 (wi ph ps))
    (syl2an2_2 (wi (wa ch ph) th))
    (syl2an2_3 (wi (wa ps th) ta)))
  (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.)
(pub theorem syl2an2r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl2an2r_1 (wi ph ps))
    (syl2an2r_2 (wi (wa ph ch) th))
    (syl2an2r_3 (wi (wa ps th) ta)))
  (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.)
(pub theorem syl2an2rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl2an2r_1 (wi ph ps))
    (syl2an2r_2 (wi (wa ph ch) th))
    (syl2an2r_3 (wi (wa ps th) ta)))
  (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.)
(pub theorem impbida ((ph wff ()) (ps wff ()) (ch wff ()))
  ((impbida_1 (wi (wa ph ps) ch))
    (impbida_2 (wi (wa ph ch) ps)))
  (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.)
(pub theorem pm3_48 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem pm3_45 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem im2anan9 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((im2an9_1 (wi ph (wi ps ch)))
    (im2an9_2 (wi th (wi ta et))))
  (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.)
(pub theorem im2anan9r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((im2an9_1 (wi ph (wi ps ch)))
    (im2an9_2 (wi th (wi ta et))))
  (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.)
(pub theorem anim12dan ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((anim12dan_1 (wi (wa ph ps) ch))
    (anim12dan_2 (wi (wa ph th) ta)))
  (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.)
(pub theorem orim12d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((orim12d_1 (wi ph (wi ps ch)))
    (orim12d_2 (wi ph (wi th ta))))
  (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.)
(pub theorem orim1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((orim1d_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem orim2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((orim1d_1 (wi ph (wi ps ch))))
  (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.)
(pub theorem orim2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_38 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_36 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_37 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_73 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_74 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem orimdi ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_76 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_75 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_8 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm2_81 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem pm2_82 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem pm2_85 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm3_2ni ((ph wff ()) (ps wff ()))
  ((pm3_2ni_1 (wn ph))
    (pm3_2ni_2 (wn ps)))
  (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.)
(pub theorem orabs ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem oranabs ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_1 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_21 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem norbi ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem nbior ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm3_43 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem jcab ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem ordi ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem ordir ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm4_76 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem andi ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem andir ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem orddi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem anddi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem pm4_39 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem pm4_38 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem bi2anan9 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((bi2an9_1 (wi ph (wb ps ch)))
    (bi2an9_2 (wi th (wb ta et))))
  (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.)
(pub theorem bi2anan9r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((bi2an9_1 (wi ph (wb ps ch)))
    (bi2an9_2 (wi th (wb ta et))))
  (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.)
(pub theorem bi2bian9 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((bi2an9_1 (wi ph (wb ps ch)))
    (bi2an9_2 (wi th (wb ta et))))
  (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.)
(pub theorem pm4_72 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem imimorb ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_33 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_36 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem bianabs ((ph wff ()) (ps wff ()) (ch wff ()))
  ((bianabs_1 (wi ph (wb ps (wa ph ch)))))
  (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.)
(pub theorem oibabs ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm3_24 ((ph wff ()))
  ()
  (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.)
(pub theorem pm2_26 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_11 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_12 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_14 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_13 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_17 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_15 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_16 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xor ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem nbi2 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xordi ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem biort ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_55 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem ornld ((ph wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem annotanannot ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem annotanannotOLD ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_21nd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((pm5_21nd_1 (wi (wa ph ps) th))
    (pm5_21nd_2 (wi (wa ph ch) th))
    (pm5_21nd_3 (wi th (wb ps ch))))
  (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.)
(pub theorem pm5_35 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_54 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem baib ((ph wff ()) (ps wff ()) (ch wff ()))
  ((baib_1 (wb ph (wa ps ch))))
  (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.)
(pub theorem baibr ((ph wff ()) (ps wff ()) (ch wff ()))
  ((baib_1 (wb ph (wa ps ch))))
  (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.)
(pub theorem rbaibr ((ph wff ()) (ps wff ()) (ch wff ()))
  ((baib_1 (wb ph (wa ps ch))))
  (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.)
(pub theorem rbaib ((ph wff ()) (ps wff ()) (ch wff ()))
  ((baib_1 (wb ph (wa ps ch))))
  (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.)
(pub theorem baibd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((baibd_1 (wi ph (wb ps (wa ch th)))))
  (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.)
(pub theorem rbaibd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((baibd_1 (wi ph (wb ps (wa ch th)))))
  (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.)
(pub theorem pm5_44 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_6 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem orcanai ((ph wff ()) (ps wff ()) (ch wff ()))
  ((orcanai_1 (wi ph (wo ps ch))))
  (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.)
(pub theorem mpbiran ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpbiran_1 ps)
    (mpbiran_2 (wb ph (wa ps ch))))
  (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.)
(pub theorem mpbiran2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpbiran2_1 ch)
    (mpbiran2_2 (wb ph (wa ps ch))))
  (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.)
(pub theorem mpbir2an ((ph wff ()) (ps wff ()) (ch wff ()))
  ((mpbir2an_1 ps)
    (mpbir2an_2 ch)
    (mpbiran2an_1 (wb ph (wa ps ch))))
  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.)
(pub theorem mpbi2and ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpbi2and_1 (wi ph ps))
    (mpbi2and_2 (wi ph ch))
    (mpbi2and_3 (wi ph (wb (wa ps ch) th))))
  (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.)
(pub theorem mpbir2and ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpbir2and_1 (wi ph ch))
    (mpbir2and_2 (wi ph th))
    (mpbir2and_3 (wi ph (wb ps (wa ch th)))))
  (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.)
(pub theorem pm5_62 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_63 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem intnan ((ph wff ()) (ps wff ()))
  ((intnan_1 (wn ph)))
  (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.)
(pub theorem intnanr ((ph wff ()) (ps wff ()))
  ((intnan_1 (wn ph)))
  (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.)
(pub theorem intnand ((ph wff ()) (ps wff ()) (ch wff ()))
  ((intnand_1 (wi ph (wn ps))))
  (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.)
(pub theorem intnanrd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((intnand_1 (wi ph (wn ps))))
  (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.)
(pub theorem niabn ((ph wff ()) (ps wff ()) (ch wff ()))
  ((niabn_1 ph))
  (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.)
(pub theorem ninba ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ninba_1 ph))
  (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.)
(pub theorem bianfi ((ph wff ()) (ps wff ()))
  ((bianfi_1 (wn ph)))
  (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.)
(pub theorem bianfd ((ph wff ()) (ps wff ()) (ch wff ()))
  ((bianfd_1 (wi ph (wn ps))))
  (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.)
(pub theorem pm4_43 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_82 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm4_83 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pclem6 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem biantr ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem orbidi ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem biluk ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_7 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem bigolden ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_71 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm5_75 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem bimsc1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem ecase2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((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)))))
  (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.)
(pub theorem ecase3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ecase3_1 (wi ph ch))
    (ecase3_2 (wi ps ch))
    (ecase3_3 (wi (wn (wo ph ps)) ch)))
  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.)
(pub theorem ecase ((ph wff ()) (ps wff ()) (ch wff ()))
  ((ecase_1 (wi (wn ph) ch))
    (ecase_2 (wi (wn ps) ch))
    (ecase_3 (wi (wa ph ps) ch)))
  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.)
(pub theorem ecase3d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((ecase3d_1 (wi ph (wi ps th)))
    (ecase3d_2 (wi ph (wi ch th)))
    (ecase3d_3 (wi ph (wi (wn (wo ps ch)) th))))
  (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.)
(pub theorem ecased ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((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))))
  (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.)
(pub theorem ecase3ad ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((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))))
  (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.)
(pub theorem ccase ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((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)))
  (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.)
(pub theorem ccased ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((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))))
  (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.)
(pub theorem ccase2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ccase2_1 (wi (wa ph ps) ta))
    (ccase2_2 (wi ch ta))
    (ccase2_3 (wi th ta)))
  (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.)
(pub theorem _4cases ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_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)))
  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.)
(pub theorem _4casesdan ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_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)))
  (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.)
(pub theorem cases ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((cases_1 (wi ph (wb ps ch)))
    (cases_2 (wi (wn ph) (wb ps th))))
  (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.)
(pub theorem dedlem0a ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem dedlem0b ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem dedlema ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem dedlemb ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cases2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cases2ALT ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem dfbi3 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem dfbi3OLD ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem pm5_24 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem _4exmid ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem _4exmidOLD ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem consensus ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm4_42 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem prlem1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((prlem1_1 (wi ph (wb et ch)))
    (prlem1_2 (wi ps (wn th))))
  (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.)
(pub theorem prlem2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem oplem1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((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))))
  (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.)
(pub theorem dn1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem bianir ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem jaoi2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((jaoi2_1 (wi (wo ph (wa (wn ph) ch)) ps)))
  (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.)
(pub theorem jaoi3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((jaoi3_1 (wi ph ps))
    (jaoi3_2 (wi (wa (wn ph) ch) ps)))
  (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 ((ph wff ()) (ps wff ()) (ch 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.)
(pub theorem dfifp2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem dfifp3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem dfifp4 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem dfifp5 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem dfifp6 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem dfifp7 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem anifp ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem ifpor ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem ifpn ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem ifptru ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem ifpfal ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem ifpid ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem casesifp ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((casesifp_1 (wi ph (wb ps ch)))
    (casesifp_2 (wi (wn ph) (wb ps th))))
  (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.)
(pub theorem ifpbi123d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((ifpbi123d_1 (wi ph (wb ps ta)))
    (ifpbi123d_2 (wi ph (wb ch et)))
    (ifpbi123d_3 (wi ph (wb th ze))))
  (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.)
(pub theorem ifpimpda ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((ifpimpda_1 (wi (wa ph ps) ch))
    (ifpimpda_2 (wi (wa ph (wn ps)) th)))
  (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.)
(pub theorem _1fpid3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_1fpid3_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem elimh ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((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))
  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.)
(pub theorem dedt ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((dedt_1 (wi (wb (wif ch ph ps) ph) (wb th ta)))
    (dedt_2 ta))
  (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.)
(pub theorem con3ALT ((ph wff ()) (ps wff ()))
  ()
  (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 ((ph wff ()) (ps wff ()) (ch wff ())) (wff ()))

--| Extend wff definition to include three-way conjunction ('and').
(term w3a ((ph wff ()) (ps wff ()) (ch 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.)
(pub theorem _3orass ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3orel1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3orrot ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3orcoma ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3orcomb ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3orcombOLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3anass ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3anan12 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3anan32 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3ancoma ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3ancomaOLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3ancomb ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3anrot ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3anan12OLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3anrotOLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3ancombOLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3anrev ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem anandi3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem anandi3r ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3anorOLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3ioran ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3ianor ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3anor ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3ianorOLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3oran ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3impa ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3impa_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem _3imp ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3imp_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem _3impOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3imp_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem _3imp31 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3imp_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem _3imp231 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3imp_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem _3imp21 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3imp_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem _3impaOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3impaOLD_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem _3impb ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3impb_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem _3impib ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3impib_1 (wi ph (wi (wa ps ch) th))))
  (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.)
(pub theorem _3impia ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3impia_1 (wi (wa ph ps) (wi ch th))))
  (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.)
(pub theorem _3impiaOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3impiaOLD_1 (wi (wa ph ps) (wi ch th))))
  (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.)
(pub theorem _3expa ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3exp_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3exp ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3exp_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3expb ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3exp_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3expia ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3exp_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3expiaOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3exp_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3expib ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3exp_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3com12 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3exp_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3com13 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3exp_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3comr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3exp_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3com23 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3exp_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3coml ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3exp_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3adant_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem _3adant2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3adant_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem _3adant3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3adant_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem _3ad2ant1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3ad2ant_1 (wi ph ch)))
  (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.)
(pub theorem _3ad2ant2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3ad2ant_1 (wi ph ch)))
  (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.)
(pub theorem _3ad2ant3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3ad2ant_1 (wi ph ch)))
  (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.)
(pub theorem simp1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem simp2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem simp3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem simp1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_3simp1i_1 (w3a ph ps ch)))
  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.)
(pub theorem simp2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_3simp1i_1 (w3a ph ps ch)))
  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.)
(pub theorem simp3i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_3simp1i_1 (w3a ph ps ch)))
  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.)
(pub theorem simp1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3simp1d_1 (wi ph (w3a ps ch th))))
  (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.)
(pub theorem simp2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3simp1d_1 (wi ph (w3a ps ch th))))
  (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.)
(pub theorem simp3d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3simp1d_1 (wi ph (w3a ps ch th))))
  (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.)
(pub theorem simp1bi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3simp1bi_1 (wb ph (w3a ps ch th))))
  (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.)
(pub theorem simp2bi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3simp1bi_1 (wb ph (w3a ps ch th))))
  (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.)
(pub theorem simp3bi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3simp1bi_1 (wb ph (w3a ps ch th))))
  (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.)
(pub theorem _3simpa ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3simpaOLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3simpb ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3simpbOLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3simpc ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3simpcOLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem simp1OLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem simp2OLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem simp3OLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3adant1OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3adantOLD_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem _3adant2OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3adantOLD_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem _3adant3OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3adantOLD_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem _3adantl1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3adantl_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem _3adantl2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3adantl_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem _3adantl3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3adantl_1 (wi (wa (wa ph ps) ch) th)))
  (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.)
(pub theorem _3adantr1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3adantr_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem _3adantr2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3adantr_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem _3adantr3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3adantr_1 (wi (wa ph (wa ps ch)) th)))
  (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.)
(pub theorem ad4ant123 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad4ant123OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad4ant124 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad4ant124OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad4ant134 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad4ant134OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad4ant234 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad4ant234OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant1l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant1lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant1r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant1rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant2l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant2r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant3l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant3r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant3r1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant3r2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3adant3r3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant3_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3ad2antl1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3ad2antl_1 (wi (wa ph ch) th)))
  (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.)
(pub theorem _3ad2antl2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3ad2antl_1 (wi (wa ph ch) th)))
  (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.)
(pub theorem _3ad2antl3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3ad2antl_1 (wi (wa ph ch) th)))
  (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.)
(pub theorem _3ad2antr1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3ad2antl_1 (wi (wa ph ch) th)))
  (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.)
(pub theorem _3ad2antr2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3ad2antl_1 (wi (wa ph ch) th)))
  (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.)
(pub theorem _3ad2antr3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3ad2antl_1 (wi (wa ph ch) th)))
  (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.)
(pub theorem ad4ant13 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad4ant13OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad4ant14 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad4ant14OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad4ant23 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad4ant23OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad4ant24 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad4ant24OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ad4ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant12 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant13 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant13OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant14 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant14OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant15 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant15OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant23 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant23OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant24 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant24OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant25 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem ad5ant25OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2_1 (wi (wa ph ps) ch)))
  (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.)
(pub theorem simpl1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpl1OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpl2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpl2OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpl3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpl3OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpr1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpr1OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpr2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpr2OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpr3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simpr3OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simp1l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simp1r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simp2l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simp2r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simp3l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simp3r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem simp11 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp12 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp13 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp21 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp22 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp23 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp31 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp32 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp33 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpll1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpll1OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpll2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpll2OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpll3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpll3OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simplr1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simplr1OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simplr2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simplr2OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simplr3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simplr3OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprl1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprl1OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprl2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprl2OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprl3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprl3OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprr1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprr1OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprr2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprr2OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprr3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simprr3OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl1l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl1lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl1r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl1rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl2l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl2lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl2r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl2rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl3l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl3lOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl3r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl3rOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpr1l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpr1r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpr2l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpr2r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpr3l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpr3r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp1ll ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp1lr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp1rl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp1rr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp2ll ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp2lr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp2rl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp2rr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp3ll ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp3lr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp3rl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simp3rr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem simpl11 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpl12 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpl13 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpl21 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpl22 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpl23 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpl31 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpl32 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpl33 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpr11 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpr12 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpr13 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpr21 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpr22 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpr23 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpr31 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpr32 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simpr33 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp1l1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp1l2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp1l3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp1r1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp1r2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp1r3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp2l1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp2l2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp2l3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp2r1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp2r2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp2r3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp3l1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp3l2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp3l3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp3r1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp3r2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp3r3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp11l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp11r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp12l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp12r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp13l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp13r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp21l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp21r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp22l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp22r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp23l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp23r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp31l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp31r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp32l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp32r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp33l ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp33r ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem simp111 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp112 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp113 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp121 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp122 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp123 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp131 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp132 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp133 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp211 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp212 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp213 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp221 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp222 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp223 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp231 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp232 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp233 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp311 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp312 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp313 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp321 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp322 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp323 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp331 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp332 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem simp333 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ()
  (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.)
(pub theorem _3anibar ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3anibar_1 (wi (w3a ph ps ch) (wb th (wa ch ta)))))
  (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.)
(pub theorem _3mix1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3mix2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3mix3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3mix1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_3mixi_1 ph))
  (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.)
(pub theorem _3mix2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_3mixi_1 ph))
  (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.)
(pub theorem _3mix3i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_3mixi_1 ph))
  (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.)
(pub theorem _3mix1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3mixd_1 (wi ph ps)))
  (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.)
(pub theorem _3mix2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3mixd_1 (wi ph ps)))
  (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.)
(pub theorem _3mix3d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3mixd_1 (wi ph ps)))
  (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.)
(pub theorem _3pm3_2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_3pm3_2i_1 ph)
    (_3pm3_2i_2 ps)
    (_3pm3_2i_3 ch))
  (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.)
(pub theorem pm3_2an3 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem pm3_2an3OLD ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem _3jca ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3jca_1 (wi ph ps))
    (_3jca_2 (wi ph ch))
    (_3jca_3 (wi ph th)))
  (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.)
(pub theorem _3jcad ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3jcad_1 (wi ph (wi ps ch)))
    (_3jcad_2 (wi ph (wi ps th)))
    (_3jcad_3 (wi ph (wi ps ta))))
  (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.)
(pub theorem mpbir3an ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpbir3an_1 ps)
    (mpbir3an_2 ch)
    (mpbir3an_3 th)
    (mpbir3an_4 (wb ph (w3a ps ch th))))
  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.)
(pub theorem mpbir3and ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((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)))))
  (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.)
(pub theorem syl3anbrc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl3anbrc_1 (wi ph ps))
    (syl3anbrc_2 (wi ph ch))
    (syl3anbrc_3 (wi ph th))
    (syl3anbrc_4 (wb ta (w3a ps ch th))))
  (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.)
(pub theorem _3anim123i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((_3anim123i_1 (wi ph ps))
    (_3anim123i_2 (wi ch th))
    (_3anim123i_3 (wi ta et)))
  (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.)
(pub theorem _3anim1i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3animi_1 (wi ph ps)))
  (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.)
(pub theorem _3anim2i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3animi_1 (wi ph ps)))
  (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.)
(pub theorem _3anim3i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3animi_1 (wi ph ps)))
  (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.)
(pub theorem _3anbi123i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((bi3_1 (wb ph ps))
    (bi3_2 (wb ch th))
    (bi3_3 (wb ta et)))
  (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.)
(pub theorem _3orbi123i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((bi3_1 (wb ph ps))
    (bi3_2 (wb ch th))
    (bi3_3 (wb ta et)))
  (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.)
(pub theorem _3anbi1i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3anbi1i_1 (wb ph ps)))
  (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.)
(pub theorem _3anbi2i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3anbi1i_1 (wb ph ps)))
  (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.)
(pub theorem _3anbi3i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3anbi1i_1 (wb ph ps)))
  (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.)
(pub theorem ex3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((ex3_1 (wi (wa (wa (wa ph ps) ch) th) ta)))
  (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.)
(pub theorem _3expOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3comOLD_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3expaOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3comOLD_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3com12OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3comOLD_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3com13OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3comOLD_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3com23OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3comOLD_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3comrOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3comOLD_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem _3imp21OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3imp21OLD_1 (wi ph (wi ps (wi ch th)))))
  (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.)
(pub theorem _3imp3i2an ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((_3imp3i2an_1 (wi (w3a ph ps ch) th))
    (_3imp3i2an_2 (wi (wa ph ch) ta))
    (_3imp3i2an_3 (wi (wa th ta) et)))
  (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.)
(pub theorem _3imp3i2anOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((_3imp3i2an_1 (wi (w3a ph ps ch) th))
    (_3imp3i2an_2 (wi (wa ph ch) ta))
    (_3imp3i2an_3 (wi (wa th ta) et)))
  (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.)
(pub theorem _3an1rsOLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3an1rsOLD_1 (wi (wa (w3a ph ps ch) th) ta)))
  (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.)
(pub theorem _3imp1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3imp1_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem _3impd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3imp1_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem _3imp2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3imp1_1 (wi ph (wi ps (wi ch (wi th ta))))))
  (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.)
(pub theorem _3exp1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3exp1_1 (wi (wa (w3a ph ps ch) th) ta)))
  (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.)
(pub theorem _3expd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3expd_1 (wi ph (wi (w3a ps ch th) ta))))
  (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.)
(pub theorem _3exp2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3exp2_1 (wi (wa ph (w3a ps ch th)) ta)))
  (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.)
(pub theorem exp5o ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((exp5o_1 (wi (w3a ph ps ch) (wi (wa th ta) et))))
  (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.)
(pub theorem exp516 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((exp516_1 (wi (wa (wa ph (w3a ps ch th)) ta) et)))
  (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.)
(pub theorem exp520 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((exp520_1 (wi (wa (w3a ph ps ch) (wa th ta)) et)))
  (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.)
(pub theorem _3impexp ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem _3an1rs ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3an1rs_1 (wi (wa (w3a ph ps ch) th) ta)))
  (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.)
(pub theorem _3anassrs ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3anassrs_1 (wi (wa ph (w3a ps ch th)) ta)))
  (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.)
(pub theorem _3an4anass ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem ad5ant245 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant245OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant234 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant234OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant235 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant235OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant123 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant123OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant124 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant124OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant125 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant125OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant134 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant134OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant135 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant135OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant145 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant145OLD ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant_1 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem ad5ant1345 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant1345_1 (wi (wa (wa (wa ph ps) ch) th) ta)))
  (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.)
(pub theorem ad5ant2345 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((ad5ant2345_1 (wi (wa (wa (wa ph ps) ch) th) ta)))
  (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.)
(pub theorem syl12anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl12anc_1 (wi ph ps))
    (syl12anc_2 (wi ph ch))
    (syl12anc_3 (wi ph th))
    (syl12anc_4 (wi (wa ps (wa ch th)) ta)))
  (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.)
(pub theorem syl21anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl12anc_1 (wi ph ps))
    (syl12anc_2 (wi ph ch))
    (syl12anc_3 (wi ph th))
    (syl21anc_4 (wi (wa (wa ps ch) th) ta)))
  (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.)
(pub theorem syl3anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl12anc_1 (wi ph ps))
    (syl12anc_2 (wi ph ch))
    (syl12anc_3 (wi ph th))
    (syl3anc_4 (wi (w3a ps ch th) ta)))
  (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.)
(pub theorem syl22anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((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)))
  (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.)
(pub theorem syl13anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((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)))
  (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.)
(pub theorem syl31anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((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)))
  (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.)
(pub theorem syl112anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((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)))
  (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.)
(pub theorem syl121anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((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)))
  (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.)
(pub theorem syl211anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((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)))
  (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.)
(pub theorem syl23anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((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)))
  (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.)
(pub theorem syl32anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((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)))
  (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.)
(pub theorem syl122anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((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)))
  (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.)
(pub theorem syl212anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((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)))
  (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.)
(pub theorem syl221anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((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)))
  (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.)
(pub theorem syl113anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((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)))
  (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.)
(pub theorem syl131anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((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)))
  (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.)
(pub theorem syl311anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((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)))
  (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.)
(pub theorem syl33anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((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)))
  (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.)
(pub theorem syl222anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((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)))
  (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.)
(pub theorem syl123anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((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)))
  (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.)
(pub theorem syl132anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((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)))
  (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.)
(pub theorem syl213anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((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)))
  (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.)
(pub theorem syl231anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((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)))
  (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.)
(pub theorem syl312anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((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)))
  (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.)
(pub theorem syl321anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((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)))
  (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.)
(pub theorem syl133anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ((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)))
  (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.)
(pub theorem syl313anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ((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)))
  (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.)
(pub theorem syl331anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ((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)))
  (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.)
(pub theorem syl223anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ((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)))
  (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.)
(pub theorem syl232anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ((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)))
  (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.)
(pub theorem syl322anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ((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)))
  (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.)
(pub theorem syl233anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()))
  ((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)))
  (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.)
(pub theorem syl323anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()))
  ((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)))
  (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.)
(pub theorem syl332anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()))
  ((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)))
  (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.)
(pub theorem syl333anc ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()) (mu wff ()) (la wff ()))
  ((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)))
  (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.)
(pub theorem syl3an1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl3an1_1 (wi ph ps))
    (syl3an1_2 (wi (w3a ps ch th) ta)))
  (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.)
(pub theorem syl3an2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl3an2_1 (wi ph ch))
    (syl3an2_2 (wi (w3a ps ch th) ta)))
  (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.)
(pub theorem syl3an3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl3an3_1 (wi ph th))
    (syl3an3_2 (wi (w3a ps ch th) ta)))
  (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.)
(pub theorem syl3an1b ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl3an1b_1 (wb ph ps))
    (syl3an1b_2 (wi (w3a ps ch th) ta)))
  (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.)
(pub theorem syl3an2b ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl3an2b_1 (wb ph ch))
    (syl3an2b_2 (wi (w3a ps ch th) ta)))
  (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.)
(pub theorem syl3an3b ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl3an3b_1 (wb ph th))
    (syl3an3b_2 (wi (w3a ps ch th) ta)))
  (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.)
(pub theorem syl3an1br ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl3an1br_1 (wb ps ph))
    (syl3an1br_2 (wi (w3a ps ch th) ta)))
  (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.)
(pub theorem syl3an2br ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl3an2br_1 (wb ch ph))
    (syl3an2br_2 (wi (w3a ps ch th) ta)))
  (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.)
(pub theorem syl3an3br ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syl3an3br_1 (wb th ph))
    (syl3an3br_2 (wi (w3a ps ch th) ta)))
  (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.)
(pub theorem syl3an ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((syl3an_1 (wi ph ps))
    (syl3an_2 (wi ch th))
    (syl3an_3 (wi ta et))
    (syl3an_4 (wi (w3a ps th et) ze)))
  (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.)
(pub theorem syl3anb ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((syl3anb_1 (wb ph ps))
    (syl3anb_2 (wb ch th))
    (syl3anb_3 (wb ta et))
    (syl3anb_4 (wi (w3a ps th et) ze)))
  (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.)
(pub theorem syl3anbr ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((syl3anbr_1 (wb ps ph))
    (syl3anbr_2 (wb th ch))
    (syl3anbr_3 (wb et ta))
    (syl3anbr_4 (wi (w3a ps th et) ze)))
  (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.)
(pub theorem syld3an3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syld3an3_1 (wi (w3a ph ps ch) th))
    (syld3an3_2 (wi (w3a ph ps th) ta)))
  (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.)
(pub theorem syld3an1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syld3an1_1 (wi (w3a ch ps th) ph))
    (syld3an1_2 (wi (w3a ph ps th) ta)))
  (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.)
(pub theorem syld3an2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((syld3an2_1 (wi (w3a ph ch th) ps))
    (syld3an2_2 (wi (w3a ph ps th) ta)))
  (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.)
(pub theorem syl3anl1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((syl3anl1_1 (wi ph ps))
    (syl3anl1_2 (wi (wa (w3a ps ch th) ta) et)))
  (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.)
(pub theorem syl3anl2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((syl3anl2_1 (wi ph ch))
    (syl3anl2_2 (wi (wa (w3a ps ch th) ta) et)))
  (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.)
(pub theorem syl3anl3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((syl3anl3_1 (wi ph th))
    (syl3anl3_2 (wi (wa (w3a ps ch th) ta) et)))
  (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.)
(pub theorem syl3anl ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()))
  ((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)))
  (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.)
(pub theorem syl3anr1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((syl3anr1_1 (wi ph ps))
    (syl3anr1_2 (wi (wa ch (w3a ps th ta)) et)))
  (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.)
(pub theorem syl3anr2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((syl3anr2_1 (wi ph th))
    (syl3anr2_2 (wi (wa ch (w3a ps th ta)) et)))
  (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.)
(pub theorem syl3anr3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((syl3anr3_1 (wi ph ta))
    (syl3anr3_2 (wi (wa ch (w3a ps th ta)) et)))
  (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.)
(pub theorem _3impdi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3impdi_1 (wi (wa (wa ph ps) (wa ph ch)) th)))
  (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.)
(pub theorem _3impdir ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3impdir_1 (wi (wa (wa ph ps) (wa ch ps)) th)))
  (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.)
(pub theorem _3anidm12 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_3anidm12_1 (wi (w3a ph ph ps) ch)))
  (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.)
(pub theorem _3anidm13 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_3anidm13_1 (wi (w3a ph ps ph) ch)))
  (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.)
(pub theorem _3anidm23 ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_3anidm23_1 (wi (w3a ph ps ps) ch)))
  (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.)
(pub theorem syl2an3an ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((syl2an3an_1 (wi ph ps))
    (syl2an3an_2 (wi ph ch))
    (syl2an3an_3 (wi th ta))
    (syl2an3an_4 (wi (w3a ps ch ta) et)))
  (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.)
(pub theorem syl2an23an ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((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)))
  (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.)
(pub theorem _3ori ((ph wff ()) (ps wff ()) (ch wff ()))
  ((_3ori_1 (w3o ph ps ch)))
  (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.)
(pub theorem _3jao ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem _3jaob ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem _3jaoi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3jaoi_1 (wi ph ps))
    (_3jaoi_2 (wi ch ps))
    (_3jaoi_3 (wi th ps)))
  (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.)
(pub theorem _3jaod ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3jaod_1 (wi ph (wi ps ch)))
    (_3jaod_2 (wi ph (wi th ch)))
    (_3jaod_3 (wi ph (wi ta ch))))
  (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.)
(pub theorem _3jaoian ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3jaoian_1 (wi (wa ph ps) ch))
    (_3jaoian_2 (wi (wa th ps) ch))
    (_3jaoian_3 (wi (wa ta ps) ch)))
  (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.)
(pub theorem _3jaodan ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3jaodan_1 (wi (wa ph ps) ch))
    (_3jaodan_2 (wi (wa ph th) ch))
    (_3jaodan_3 (wi (wa ph ta) ch)))
  (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.)
(pub theorem mpjao3dan ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((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))))
  (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.)
(pub theorem _3jaao ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((_3jaao_1 (wi ph (wi ps ch)))
    (_3jaao_2 (wi th (wi ta ch)))
    (_3jaao_3 (wi et (wi ze ch))))
  (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.)
(pub theorem syl3an9b ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((syl3an9b_1 (wi ph (wb ps ch)))
    (syl3an9b_2 (wi th (wb ch ta)))
    (syl3an9b_3 (wi et (wb ta ze))))
  (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.)
(pub theorem _3orbi123d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((bi3d_1 (wi ph (wb ps ch)))
    (bi3d_2 (wi ph (wb th ta)))
    (bi3d_3 (wi ph (wb et ze))))
  (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.)
(pub theorem _3anbi123d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((bi3d_1 (wi ph (wb ps ch)))
    (bi3d_2 (wi ph (wb th ta)))
    (bi3d_3 (wi ph (wb et ze))))
  (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.)
(pub theorem _3anbi12d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((_3anbi12d_1 (wi ph (wb ps ch)))
    (_3anbi12d_2 (wi ph (wb th ta))))
  (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.)
(pub theorem _3anbi13d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((_3anbi12d_1 (wi ph (wb ps ch)))
    (_3anbi12d_2 (wi ph (wb th ta))))
  (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.)
(pub theorem _3anbi23d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((_3anbi12d_1 (wi ph (wb ps ch)))
    (_3anbi12d_2 (wi ph (wb th ta))))
  (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.)
(pub theorem _3anbi1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3anbi1d_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem _3anbi2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3anbi1d_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem _3anbi3d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3anbi1d_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem _3anim123d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((_3anim123d_1 (wi ph (wi ps ch)))
    (_3anim123d_2 (wi ph (wi th ta)))
    (_3anim123d_3 (wi ph (wi et ze))))
  (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.)
(pub theorem _3orim123d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((_3anim123d_1 (wi ph (wi ps ch)))
    (_3anim123d_2 (wi ph (wi th ta)))
    (_3anim123d_3 (wi ph (wi et ze))))
  (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.)
(pub theorem an6 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem _3an6 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem _3or6 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ()
  (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.)
(pub theorem mp3an1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mp3an1_1 ph)
    (mp3an1_2 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem mp3an2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mp3an2_1 ps)
    (mp3an2_2 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem mp3an3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mp3an3_1 ch)
    (mp3an3_2 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem mp3an12 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mp3an12_1 ph)
    (mp3an12_2 ps)
    (mp3an12_3 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem mp3an13 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mp3an13_1 ph)
    (mp3an13_2 ch)
    (mp3an13_3 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem mp3an23 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mp3an23_1 ps)
    (mp3an23_2 ch)
    (mp3an23_3 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem mp3an1i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mp3an1i_1 ps)
    (mp3an1i_2 (wi ph (wi (w3a ps ch th) ta))))
  (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.)
(pub theorem mp3anl1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mp3anl1_1 ph)
    (mp3anl1_2 (wi (wa (w3a ph ps ch) th) ta)))
  (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.)
(pub theorem mp3anl2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mp3anl2_1 ps)
    (mp3anl2_2 (wi (wa (w3a ph ps ch) th) ta)))
  (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.)
(pub theorem mp3anl3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mp3anl3_1 ch)
    (mp3anl3_2 (wi (wa (w3a ph ps ch) th) ta)))
  (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.)
(pub theorem mp3anr1 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mp3anr1_1 ps)
    (mp3anr1_2 (wi (wa ph (w3a ps ch th)) ta)))
  (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.)
(pub theorem mp3anr2 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mp3anr2_1 ch)
    (mp3anr2_2 (wi (wa ph (w3a ps ch th)) ta)))
  (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.)
(pub theorem mp3anr3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mp3anr3_1 th)
    (mp3anr3_2 (wi (wa ph (w3a ps ch th)) ta)))
  (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.)
(pub theorem mp3an ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mp3an_1 ph)
    (mp3an_2 ps)
    (mp3an_3 ch)
    (mp3an_4 (wi (w3a ph ps ch) th)))
  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.)
(pub theorem mpd3an3 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpd3an3_2 (wi (wa ph ps) ch))
    (mpd3an3_3 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem mpd3an23 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((mpd3an23_1 (wi ph ps))
    (mpd3an23_2 (wi ph ch))
    (mpd3an23_3 (wi (w3a ph ps ch) th)))
  (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.)
(pub theorem mp3and ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((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))))
  (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.)
(pub theorem mp3an12i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mp3an12i_1 ph)
    (mp3an12i_2 ps)
    (mp3an12i_3 (wi ch th))
    (mp3an12i_4 (wi (w3a ph ps th) ta)))
  (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.)
(pub theorem mp3an2i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((mp3an2i_1 ph)
    (mp3an2i_2 (wi ps ch))
    (mp3an2i_3 (wi ps th))
    (mp3an2i_4 (wi (w3a ph ch th) ta)))
  (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.)
(pub theorem mp3an3an ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((mp3an3an_1 ph)
    (mp3an3an_2 (wi ps ch))
    (mp3an3an_3 (wi th ta))
    (mp3an3an_4 (wi (w3a ph ch ta) et)))
  (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.)
(pub theorem mp3an2ani ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((mp3an2ani_1 ph)
    (mp3an2ani_2 (wi ps ch))
    (mp3an2ani_3 (wi (wa ps th) ta))
    (mp3an2ani_4 (wi (w3a ph ch ta) et)))
  (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.)
(pub theorem biimp3a ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((biimp3a_1 (wi (wa ph ps) (wb ch th))))
  (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.)
(pub theorem biimp3ar ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((biimp3a_1 (wi (wa ph ps) (wb ch th))))
  (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.)
(pub theorem _3anandis ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3anandis_1 (wi (w3a (wa ph ps) (wa ph ch) (wa ph th)) ta)))
  (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.)
(pub theorem _3anandirs ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3anandirs_1 (wi (w3a (wa ph th) (wa ps th) (wa ch th)) ta)))
  (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.)
(pub theorem ecase23d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((ecase23d_1 (wi ph (wn ch)))
    (ecase23d_2 (wi ph (wn th)))
    (ecase23d_3 (wi ph (w3o ps ch th))))
  (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.)
(pub theorem _3ecase ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_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)))
  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.)
(pub theorem _3bior1fd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3biorfd_1 (wi ph (wn th))))
  (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.)
(pub theorem _3bior1fand ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((_3biorfd_1 (wi ph (wn th))))
  (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.)
(pub theorem _3bior2fd ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3biorfd_1 (wi ph (wn th)))
    (_3biorfd_2 (wi ph (wn ch))))
  (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.)
(pub theorem _3biant1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((_3biantd_1 (wi ph th)))
  (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.)
(pub theorem intn3an1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((intn3and_1 (wi ph (wn ps))))
  (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.)
(pub theorem intn3an2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((intn3and_1 (wi ph (wn ps))))
  (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.)
(pub theorem intn3an3d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((intn3and_1 (wi ph (wn ps))))
  (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.)
(pub theorem an3andi ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem an33rean ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()) (si wff ()) (rh wff ()))
  ()
  (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 ((ph wff ()) (ps 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.)
(pub theorem nanan ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem nancom ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem nannan ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem nanim ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem nannot ((ps wff ()))
  ()
  (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.)
(pub theorem nanbi ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem nanbi1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem nanbi2 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem nanbi12 ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ()
  (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.)
(pub theorem nanbi1i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((nanbii_1 (wb ph ps)))
  (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.)
(pub theorem nanbi2i ((ph wff ()) (ps wff ()) (ch wff ()))
  ((nanbii_1 (wb ph ps)))
  (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.)
(pub theorem nanbi12i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((nanbii_1 (wb ph ps))
    (nanbi12i_2 (wb ch th)))
  (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.)
(pub theorem nanbi1d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((nanbid_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem nanbi2d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((nanbid_1 (wi ph (wb ps ch))))
  (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.)
(pub theorem nanbi12d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((nanbid_1 (wi ph (wb ps ch)))
    (nanbi12d_2 (wi ph (wb th ta))))
  (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 ((ph wff ()) (ps 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.)
(pub theorem xnor ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xorcom ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xorass ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem excxor ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xor2 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xoror ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xornan ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xornan2 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xorneg2 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xorneg1 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xorneg ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem xorbi12i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()))
  ((xorbi12_1 (wb ph ps))
    (xorbi12_2 (wb ch th)))
  (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.)
(pub theorem xorbi12d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ((xor12d_1 (wi ph (wb ps ch)))
    (xor12d_2 (wi ph (wb th ta))))
  (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.)
(pub theorem anxordi ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem xorexmid ((ph wff ()))
  ()
  (wxo ph (wn ph))
  ()
  (mpbir ((wxo ph (wn ph)) (wn (wb ph (wn ph))))
    (pm5_19 (ph))
    (df_xor (ph (wn ph)))))

(sort setvar pure)

--| 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 ((x setvar) (ph wff (x))) (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 ((x1 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 ((A class ()) (B 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.)
(pub theorem trujust ((x2 setvar) (y setvar))
  ()
  (wb (wi (wal x2 (wceq (cv x2) (cv x2))) (wal x2 (wceq (cv x2) (cv x2)))) (wi (wal y (wceq (cv y) (cv y))) (wal y (wceq (cv y) (cv y)))))
  ()
  (_2th ((wi (wal x2 (wceq (cv x2) (cv x2))) (wal x2 (wceq (cv x2) (cv x2)))) (wi (wal y (wceq (cv y) (cv y))) (wal y (wceq (cv y) (cv y)))))
    (id ((wal x2 (wceq (cv x2) (cv x2)))))
    (id ((wal y (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 ((x2 setvar))
  ()
  (wb (wtru) (wi (wal x2 (wceq (cv x2) (cv x2))) (wal x2 (wceq (cv x2) (cv x2))))))

--| The truth value ` T. ` is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
(pub theorem tru ()
  ()
  (wtru)
  ((x2 setvar))
  (mpbir ((wtru) (wi (wal x2 (wceq (cv x2) (cv x2))) (wal x2 (wceq (cv x2) (cv x2)))))
    (id ((wal x2 (wceq (cv x2) (cv x2)))))
    (df_tru (x2))))

--| 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.)
(pub theorem fal ()
  ()
  (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.)
(pub theorem dftru2 ((ph wff ()))
  ()
  (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.)
(pub theorem trut ((ph wff ()))
  ()
  (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.)
(pub theorem trud ((ph wff ()))
  ((trud_1 (wi (wtru) ph)))
  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.)
(pub theorem tbtru ((ph wff ()))
  ()
  (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.)
(pub theorem nbfal ((ph wff ()))
  ()
  (wb (wn ph) (wb ph (wfal)))
  ()
  (nbn ((wfal) ph)
    (fal ())))

--| A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
(pub theorem bitru ((ph wff ()))
  ((bitru_1 ph))
  (wb ph (wtru))
  ()
  (_2th (ph (wtru))
    bitru_1
    (tru ())))

--| A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
(pub theorem bifal ((ph wff ()))
  ((bifal_1 (wn ph)))
  (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.)
(pub theorem falim ((ph wff ()))
  ()
  (wi (wfal) ph)
  ()
  (pm2_21i ((wfal) ph)
    (fal ())))

--| The truth value ` F. ` implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
(pub theorem falimd ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem a1tru ((ph wff ()))
  ()
  (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.)
(pub theorem truan ((ph wff ()))
  ()
  (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.)
(pub theorem dfnot ((ph wff ()))
  ()
  (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.)
(pub theorem inegd ((ph wff ()) (ps wff ()))
  ((inegd_1 (wi (wa ph ps) (wfal))))
  (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.)
(pub theorem efald ((ph wff ()) (ps wff ()))
  ((efald_1 (wi (wa ph (wn ps)) (wfal))))
  (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.)
(pub theorem pm2_21fal ((ph wff ()) (ps wff ()))
  ((pm2_21fal_1 (wi ph ps))
    (pm2_21fal_2 (wi ph (wn ps))))
  (wi ph (wfal))
  ()
  (pm2_21dd (ph ps (wfal))
    pm2_21fal_1
    pm2_21fal_2))

--| A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(pub theorem truantru ()
  ()
  (wb (wa (wtru) (wtru)) (wtru))
  ()
  (anidm ((wtru))))

--| A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(pub theorem truanfal ()
  ()
  (wb (wa (wtru) (wfal)) (wfal))
  ()
  (truan ((wfal))))

--| A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(pub theorem falantru ()
  ()
  (wb (wa (wfal) (wtru)) (wfal))
  ()
  (bifal ((wa (wfal) (wtru)))
    (intnanr ((wfal) (wtru))
      (fal ()))))

--| A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(pub theorem falanfal ()
  ()
  (wb (wa (wfal) (wfal)) (wfal))
  ()
  (anidm ((wfal))))

--| A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(pub theorem truortru ()
  ()
  (wb (wo (wtru) (wtru)) (wtru))
  ()
  (oridm ((wtru))))

--| A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(pub theorem truorfal ()
  ()
  (wb (wo (wtru) (wfal)) (wtru))
  ()
  (bitru ((wo (wtru) (wfal)))
    (orci ((wtru) (wfal))
      (tru ()))))

--| A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(pub theorem falortru ()
  ()
  (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.)
(pub theorem falorfal ()
  ()
  (wb (wo (wfal) (wfal)) (wfal))
  ()
  (oridm ((wfal))))

--| A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(pub theorem truimtru ()
  ()
  (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.)
(pub theorem truimfal ()
  ()
  (wb (wi (wtru) (wfal)) (wfal))
  ()
  (bicomi ((wfal) (wi (wtru) (wfal)))
    (trut ((wfal)))))

--| A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(pub theorem falimtru ()
  ()
  (wb (wi (wfal) (wtru)) (wtru))
  ()
  (bitru ((wi (wfal) (wtru)))
    (falim ((wtru)))))

--| A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(pub theorem falimfal ()
  ()
  (wb (wi (wfal) (wfal)) (wtru))
  ()
  (bitru ((wi (wfal) (wfal)))
    (id ((wfal)))))

--| A ` -. ` identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(pub theorem nottru ()
  ()
  (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.)
(pub theorem notfal ()
  ()
  (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.)
(pub theorem trubitru ()
  ()
  (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.)
(pub theorem falbitru ()
  ()
  (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.)
(pub theorem trubifal ()
  ()
  (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.)
(pub theorem falbifal ()
  ()
  (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.)
(pub theorem trunantru ()
  ()
  (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.)
(pub theorem trunanfal ()
  ()
  (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.)
(pub theorem falnantru ()
  ()
  (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.)
(pub theorem falnanfal ()
  ()
  (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.)
(pub theorem truxortru ()
  ()
  (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.)
(pub theorem truxorfal ()
  ()
  (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.)
(pub theorem falxortru ()
  ()
  (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.)
(pub theorem falxorfal ()
  ()
  (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 ((ph wff ()) (ps wff ()) (ch 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.)
(pub theorem hadbi123d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((hadbid_1 (wi ph (wb ps ch)))
    (hadbid_2 (wi ph (wb th ta)))
    (hadbid_3 (wi ph (wb et ze))))
  (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.)
(pub theorem hadbi123i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((hadbii_1 (wb ph ps))
    (hadbii_2 (wb ch th))
    (hadbii_3 (wb ta et)))
  (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.)
(pub theorem hadass ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem hadbi ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem hadcoma ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem hadcomb ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem hadrot ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem hadnot ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem had1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem had0 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem hadifp ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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 ((ph wff ()) (ps wff ()) (ch 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.)
(pub theorem cador ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cadan ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cadbi123d ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()) (ze wff ()))
  ((cadbid_1 (wi ph (wb ps ch)))
    (cadbid_2 (wi ph (wb th ta)))
    (cadbid_3 (wi ph (wb et ze))))
  (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.)
(pub theorem cadbi123i ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()) (et wff ()))
  ((cadbii_1 (wb ph ps))
    (cadbii_2 (wb ch th))
    (cadbii_3 (wb ta et)))
  (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.)
(pub theorem cadcoma ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cadcomb ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cadrot ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cadnot ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cad1 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cad0 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cadifp ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cad11 ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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.)
(pub theorem cadtru ((ph wff ()))
  ()
  (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.)
(pub theorem minimp ((ph wff ()) (ps wff ()) (ch wff ()) (th wff ()) (ta wff ()))
  ()
  (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.)
(pub theorem minimp_sylsimp ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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 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 (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 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 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)))))
        (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 (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 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))))))
        (minimp ((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 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 (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 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 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 ph ps) (wi (wi ph ps) ch))))
      (mp2 ((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 (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 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 (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 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 (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 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 (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 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 (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 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 (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 (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 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 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 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 ph ps) (wi ph ps)) (wi (wi ph ps) (wi ph ps))) (wi (wi ph ps) (wi ph ps)))) (wi (wi ph ps) ch))))
        (minimp ((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 ph ps) (wi ph ps) (wi (wi ph ps) ch) (wi ph ps)))
        (minimp ((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 (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 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 (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 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 (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 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 (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 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 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 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) (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)))))
    (mp2b ((wi (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 ph ps) (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 (wi (wi (wi ph ps) ch) (wi ph ps)) (wi (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 (wi ph ps) ch) ch))) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)))) (wi (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))))
      (minimp ((wi ph ps) (wi ph ps) (wi ph ps) (wi ph ps) (wi ph ps)))
      (minimp ((wi (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 ph ps) (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) ch) (wi (wi (wi ph ps) ch) ch)))
      (mp2 ((wi (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 (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 (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 (wi (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 (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 (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 (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 (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 (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 (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 (wi (wi ph ps) ch) (wi ph ps)) (wi (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 (wi ph ps) ch) ch)))) (wi (wi (wi (wi ph ps) (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 (wi (wi (wi ph ps) ch) (wi ph ps)) (wi (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 (wi ph ps) ch) ch))) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)))) (wi (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)))))
        (minimp ((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 (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 (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 (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 (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 (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) ch) (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)))) ch))
        (minimp ((wi (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 (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 (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 (wi (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 (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 (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 (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 (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 (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 (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 (wi (wi ph ps) ch) (wi ph ps)) (wi (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 (wi ph ps) ch) ch))) (wi ph ps) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch))))))
    (mp2b ((wi (wi (wi ph ps) ch) (wi (wi (wi (wi ph ps) ch) (wi (wi ph ps) ch)) (wi (wi (wi (wi (wi ph ps) ch) (wi (wi ph ps) ch)) (wi (wi (wi ph ps) ch) (wi (wi ph ps) ch))) (wi (wi (wi ph ps) ch) (wi (wi ph ps) ch))))) (wi (wi (wi (wi ph ps) ch) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch))) (wi (wi (wi ps (wi (wi ph ps) ch)) (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi ps ch))) (wi (wi (wi ph ps) ch) (wi ps ch)))) (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi (wi ph ps) ch) (wi ps ch))))
      (minimp ((wi (wi ph ps) ch) (wi (wi ph ps) ch) (wi (wi ph ps) ch) (wi (wi ph ps) ch) (wi (wi ph ps) ch)))
      (minimp ((wi (wi (wi ph ps) ch) (wi (wi (wi (wi ph ps) ch) (wi (wi ph ps) ch)) (wi (wi (wi (wi (wi ph ps) ch) (wi (wi ph ps) ch)) (wi (wi (wi ph ps) ch) (wi (wi ph ps) ch))) (wi (wi (wi ph ps) ch) (wi (wi ph ps) ch))))) (wi (wi ph ps) ch) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) ps (wi ps ch)))
      (mp2 ((wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch))) (wi (wi (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch))) (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)))) (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)))))) (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ps (wi (wi ph ps) ch)) (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi ps ch)))) (wi (wi (wi (wi (wi ph ps) ch) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch))) (wi (wi (wi ps (wi (wi ph ps) ch)) (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi ps ch))) (wi (wi (wi ph ps) ch) (wi ps ch)))) (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi (wi ph ps) ch) (wi ps ch)))))
        (minimp ((wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch))))
        (minimp ((wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) ps (wi (wi ph ps) ch) ph ch))
        (minimp ((wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch))) (wi (wi (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch))) (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)))) (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)))))) (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi (wi ps (wi (wi ph ps) ch)) (wi (wi (wi ph ps) (wi (wi (wi ph ps) ch) ch)) (wi ps ch))) (wi (wi ph ps) ch) (wi (wi (wi ph ps) ch) (wi ps ch))))))))

--| Derivation of ~ ax-1 from ~ ax-mp and ~ minimp . (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(pub theorem minimp_ax1 ((ph wff ()) (ps wff ()))
  ()
  (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.)
(pub theorem minimp_ax2c ((ph wff ()) (ps wff ()) (ch wff ()))
  ()
  (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 (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) ph ps (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ch)))
    (ax_mp ((wi (wi (wi (wi ph (wi ps ch)) (wi ph (wi ps ch))) (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 (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))))
      (mp2 ((wi (wi ph (wi ps ch)) (wi (wi (wi ph (wi ps ch)) (wi ph (wi ps ch))) (wi (wi (wi (wi ph (wi ps ch)) (wi ph (wi ps ch))) (wi (wi ph (wi ps ch)) (wi ph (wi ps ch)))) (wi (wi ph (wi ps ch)) (wi ph (wi ps ch)))))) (wi (wi ph (wi ps ch)) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi ps ch))) (wi (wi (wi (wi ph (wi ps ch)) (wi ph (wi ps ch))) (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))))
        (minimp ((wi ph (wi ps ch)) (wi ph (wi ps ch)) (wi ph (wi ps ch)) (wi ph (wi ps ch)) (wi ph (wi ps ch))))
        (ax_mp ((wi (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi ph (wi ps ch))) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi ps ch))) (wi (wi ph (wi ps ch)) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi ps ch))))
          (mp2 ((wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph))) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph))))) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) ph) (wi (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi ph (wi ps ch))) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi ps ch))))
            (minimp ((wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)))
            (mp2 ((wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) (wi (wi (wi ph (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) ph))
              (minimp ((wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) ph ph ph ph))
              (mp2 ((wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))))) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))))))) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) (wi (wi (wi ph (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)))
                (minimp ((wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))))
                (minimp ((wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph ph ph ph))
                (minimp ((wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))))) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))))))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph ph)))
              (ax_mp ((wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph))) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph))))) (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) (wi (wi (wi (wi ph (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) ph))))
                (minimp ((wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)))
                (minimp ((wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph))) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph))))) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) (wi ph (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph)))) ph))))
            (minimp ((wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph)) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph))) (wi (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph))))) (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) ph (wi (wi (wi ph ph) (wi (wi (wi ph ph) (wi ph ph)) (wi ph ph))) ph) (wi ps ch))))
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    (mp2 ((wi (wi (wi (wi (wi ph (wi ps ch)) (wi ph ch)) (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 ph ps) (wi ph ps)) (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)))) (wi (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)))))) (wi (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 (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)))))) (wi (wi (wi (wi (wi ph ps) (wi ph ps)) (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)))) (wi (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))))) (wi (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 (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))))))
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        (ax_mp ((wi (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 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 ph ps) (wi ph ps)) (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))))))
          (minimp ((wi ph ps) (wi ph ps) (wi ph ps) (wi ph ps) (wi ph ps)))
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        (minimp ((wi (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 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 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 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 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 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 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 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 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 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 ph ps) (wi ph ps)) (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)))) (wi (wi ph (wi ps ch)) (wi ph ch)) (wi (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)))))))
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      (minimp_sylsimp ((wi (wi (wi ph (wi ps ch)) (wi ph ch)) (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 ph ps) (wi ph ps)) (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)))) (wi (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))))) (wi (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 (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))))))))))