%   ORIGINAL: 'h4/thm/pair/PEXISTS_THM_'
% 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. (?x y. P x y) <=> $exists ('h4/const/pair/UNCURRY' (\x y. P x y))
%   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 y. happ (happ (happ _1 P) x) y <=> happ (happ P x) y) ==> (!_0. (!P x. happ (happ _0 P) x = happ (happ _1 P) x) ==> (!P. (?x y. happ (happ P x) y) <=> $exists ('h4/const/pair/UNCURRY' (happ _0 P))))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f61311,V_3f61307]: ![F, G]: (![X]: s(V_3f61307,happ(s(fun(V_3f61311,V_3f61307),F),s(V_3f61311,X))) = s(V_3f61307,happ(s(fun(V_3f61311,V_3f61307),G),s(V_3f61311,X))) => s(fun(V_3f61311,V_3f61307),F) = s(fun(V_3f61311,V_3f61307),G))).
fof('h4/thm/pair/PEXISTS_THM_', conjecture, ![A,B]: ![V__1]: (![P, X, Y]: s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),V__1),s(fun(A,fun(B,bool)),P))),s(A,X))),s(B,Y))) = s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),P),s(A,X))),s(B,Y))) => ![V__0]: (![P, X]: s(fun(B,bool),happ(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),V__0),s(fun(A,fun(B,bool)),P))),s(A,X))) = s(fun(B,bool),happ(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),V__1),s(fun(A,fun(B,bool)),P))),s(A,X))) => ![P]: (?[X, Y]: p(s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),P),s(A,X))),s(B,Y)))) <=> p(s(bool,'$exists'(s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/pair/UNCURRY'(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),V__0),s(fun(A,fun(B,bool)),P)))))))))))).
