%   ORIGINAL: 'h4/thm/pair/EXISTS_PROD_'
% 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. (?p. P p) <=> (?p_1 p_2. P ('h4/const/pair/,' p_1 p_2))
%   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: !P. (?p. happ P p) <=> (?p_1 p_2. happ P ('h4/const/pair/,' p_1 p_2))
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_3f61161,V_3f61157]: ![F, G]: (![X]: s(V_3f61157,happ(s(fun(V_3f61161,V_3f61157),F),s(V_3f61161,X))) = s(V_3f61157,happ(s(fun(V_3f61161,V_3f61157),G),s(V_3f61161,X))) => s(fun(V_3f61161,V_3f61157),F) = s(fun(V_3f61161,V_3f61157),G))).
fof('h4/thm/pair/EXISTS_PROD_', conjecture, ![A,B]: ![P]: (?[P0]: p(s(bool,happ(s(fun('h4/type/pair/prod'(A,B),bool),P),s('h4/type/pair/prod'(A,B),P0)))) <=> ?[P__1, P__2]: p(s(bool,happ(s(fun('h4/type/pair/prod'(A,B),bool),P),s('h4/type/pair/prod'(A,B),'h4/const/pair/,'(s(A,P__1),s(B,P__2)))))))).
