%   ORIGINAL: 'h4/thm/quotient/EXISTS_REGULAR_'
% 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. (!x. P x ==> Q x) ==> $exists P ==> $exists Q
%   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 Q. (!x. happ P x ==> happ Q x) ==> $exists P ==> $exists Q
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f76390,V_3f76386]: ![F, G]: (![X]: s(V_3f76386,happ(s(fun(V_3f76390,V_3f76386),F),s(V_3f76390,X))) = s(V_3f76386,happ(s(fun(V_3f76390,V_3f76386),G),s(V_3f76390,X))) => s(fun(V_3f76390,V_3f76386),F) = s(fun(V_3f76390,V_3f76386),G))).
fof('h4/thm/quotient/EXISTS_REGULAR_', conjecture, ![A]: ![P, Q]: (![X]: (p(s(bool,happ(s(fun(A,bool),P),s(A,X)))) => p(s(bool,happ(s(fun(A,bool),Q),s(A,X))))) => (p(s(bool,'$exists'(s(fun(A,bool),P)))) => p(s(bool,'$exists'(s(fun(A,bool),Q))))))).
