%   ORIGINAL: 'h4/thm/bool/EXISTS_UNIQUE_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. 'h4/const/bool/?!' (\x. P x) <=> (?x. P x) /\ (!x y. P x /\ P y ==> 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: !_0. (!P x. happ (happ _0 P) x <=> happ P x) ==> (!P. 'h4/const/bool/?!' (happ _0 P) <=> (?x. happ P x) /\ (!x y. happ P x /\ happ P y ==> x = y))
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_3f34328,V_3f34324]: ![F, G]: (![X]: s(V_3f34324,happ(s(fun(V_3f34328,V_3f34324),F),s(V_3f34328,X))) = s(V_3f34324,happ(s(fun(V_3f34328,V_3f34324),G),s(V_3f34328,X))) => s(fun(V_3f34328,V_3f34324),F) = s(fun(V_3f34328,V_3f34324),G))).
fof('h4/thm/bool/EXISTS_UNIQUE_THM_', conjecture, ![A]: ![V__0]: (![P, X]: 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,X))) = s(bool,happ(s(fun(A,bool),P),s(A,X))) => ![P]: (p(s(bool,'h4/const/bool/?!'(s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),V__0),s(fun(A,bool),P)))))) <=> (?[X]: p(s(bool,happ(s(fun(A,bool),P),s(A,X)))) & ![X, Y]: ((p(s(bool,happ(s(fun(A,bool),P),s(A,X)))) & p(s(bool,happ(s(fun(A,bool),P),s(A,Y))))) => s(A,X) = s(A,Y)))))).
