%   ORIGINAL: 'h4/thm/pred_set/GSPEC_AND_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !P Q. 'h4/const/pred_set/GSPEC' (\x. 'h4/const/pair/,' x (P x /\ Q x)) = 'h4/const/pred_set/INTER' ('h4/const/pred_set/GSPEC' (\x. 'h4/const/pair/,' x (P x))) ('h4/const/pred_set/GSPEC' (\x. 'h4/const/pair/,' x (Q x)))
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !_1. (!P x. happ (happ _1 P) x = 'h4/const/pair/,' x (happ P x)) ==> (!_0. (!P Q x. ?v. (v <=> happ P x /\ happ Q x) /\ happ (happ (happ _0 P) Q) x = 'h4/const/pair/,' x v) ==> (!P Q. 'h4/const/pred_set/GSPEC' (happ (happ _0 P) Q) = 'h4/const/pred_set/INTER' ('h4/const/pred_set/GSPEC' (happ _1 P)) ('h4/const/pred_set/GSPEC' (happ _1 Q))))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f23650,V_3f23646]: ![F, G]: (![X]: s(V_3f23646,happ(s(fun(V_3f23650,V_3f23646),F),s(V_3f23650,X))) = s(V_3f23646,happ(s(fun(V_3f23650,V_3f23646),G),s(V_3f23650,X))) => s(fun(V_3f23650,V_3f23646),F) = s(fun(V_3f23650,V_3f23646),G))).
fof('h4/thm/pred_set/GSPEC_AND_', conjecture, ![A]: ![V__1]: (![P, X]: s('h4/type/pair/prod'(A,bool),happ(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(fun(A,bool),fun(A,'h4/type/pair/prod'(A,bool))),V__1),s(fun(A,bool),P))),s(A,X))) = s('h4/type/pair/prod'(A,bool),'h4/const/pair/,'(s(A,X),s(bool,happ(s(fun(A,bool),P),s(A,X))))) => ![V__0]: (![P, Q, X]: ?[V]: ((p(s(bool,V)) <=> (p(s(bool,happ(s(fun(A,bool),P),s(A,X)))) & p(s(bool,happ(s(fun(A,bool),Q),s(A,X)))))) & s('h4/type/pair/prod'(A,bool),happ(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(fun(A,bool),fun(A,'h4/type/pair/prod'(A,bool))),happ(s(fun(fun(A,bool),fun(fun(A,bool),fun(A,'h4/type/pair/prod'(A,bool)))),V__0),s(fun(A,bool),P))),s(fun(A,bool),Q))),s(A,X))) = s('h4/type/pair/prod'(A,bool),'h4/const/pair/,'(s(A,X),s(bool,V)))) => ![P, Q]: s(fun(A,bool),'h4/const/pred_set/GSPEC'(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(fun(A,bool),fun(A,'h4/type/pair/prod'(A,bool))),happ(s(fun(fun(A,bool),fun(fun(A,bool),fun(A,'h4/type/pair/prod'(A,bool)))),V__0),s(fun(A,bool),P))),s(fun(A,bool),Q))))) = s(fun(A,bool),'h4/const/pred_set/INTER'(s(fun(A,bool),'h4/const/pred_set/GSPEC'(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(fun(A,bool),fun(A,'h4/type/pair/prod'(A,bool))),V__1),s(fun(A,bool),P))))),s(fun(A,bool),'h4/const/pred_set/GSPEC'(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(fun(A,bool),fun(A,'h4/type/pair/prod'(A,bool))),V__1),s(fun(A,bool),Q)))))))))).
