%   ORIGINAL: 'h4/thm/res_quan/RES_EXISTS_EQUAL_'
% 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 j. 'h4/const/bool/RES_EXISTS' ($equals j) (\i. P i) <=> P j
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !_0. (!P i. happ (happ _0 P) i <=> happ P i) ==> (!P j. 'h4/const/bool/RES_EXISTS' ($equals j) (happ _0 P) <=> happ P j)
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_3f31028,V_3f31024]: ![F, G]: (![X]: s(V_3f31024,happ(s(fun(V_3f31028,V_3f31024),F),s(V_3f31028,X))) = s(V_3f31024,happ(s(fun(V_3f31028,V_3f31024),G),s(V_3f31028,X))) => s(fun(V_3f31028,V_3f31024),F) = s(fun(V_3f31028,V_3f31024),G))).
fof('h4/thm/res_quan/RES_EXISTS_EQUAL_', conjecture, ![A]: ![V__0]: (![P, I]: s(bool,happ(s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),V__0),s(fun(A,bool),P))),s(A,I))) = s(bool,happ(s(fun(A,bool),P),s(A,I))) => ![P, J]: s(bool,'h4/const/bool/RES_EXISTS'(s(fun(A,bool),'$equals'(s(A,J))),s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),V__0),s(fun(A,bool),P))))) = s(bool,happ(s(fun(A,bool),P),s(A,J))))).
