%   ORIGINAL: 'h4/thm/quotient/EXISTS_UNIQUE_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 E Q. (!x. P x ==> 'h4/const/quotient/respects' E x /\ Q x) /\ (!x y. 'h4/const/quotient/respects' E x /\ Q x /\ 'h4/const/quotient/respects' E y /\ Q y ==> E x y) ==> 'h4/const/bool/?!' P ==> 'h4/const/quotient/RES_EXISTS_EQUIV' E 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 E Q. (!x. happ P x ==> 'h4/const/quotient/respects' E x /\ happ Q x) /\ (!x y. 'h4/const/quotient/respects' E x /\ happ Q x /\ 'h4/const/quotient/respects' E y /\ happ Q y ==> happ (happ E x) y) ==> 'h4/const/bool/?!' P ==> 'h4/const/quotient/RES_EXISTS_EQUIV' E 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_3f76565,V_3f76561]: ![F, G]: (![X]: s(V_3f76561,happ(s(fun(V_3f76565,V_3f76561),F),s(V_3f76565,X))) = s(V_3f76561,happ(s(fun(V_3f76565,V_3f76561),G),s(V_3f76565,X))) => s(fun(V_3f76565,V_3f76561),F) = s(fun(V_3f76565,V_3f76561),G))).
fof('h4/thm/quotient/EXISTS_UNIQUE_REGULAR_', conjecture, ![A]: ![P, E, Q]: ((![X]: (p(s(bool,happ(s(fun(A,bool),P),s(A,X)))) => (p(s(bool,'h4/const/quotient/respects'(s(fun(A,fun(A,bool)),E),s(A,X)))) & p(s(bool,happ(s(fun(A,bool),Q),s(A,X)))))) & ![X, Y]: ((p(s(bool,'h4/const/quotient/respects'(s(fun(A,fun(A,bool)),E),s(A,X)))) & (p(s(bool,happ(s(fun(A,bool),Q),s(A,X)))) & (p(s(bool,'h4/const/quotient/respects'(s(fun(A,fun(A,bool)),E),s(A,Y)))) & p(s(bool,happ(s(fun(A,bool),Q),s(A,Y))))))) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),E),s(A,X))),s(A,Y)))))) => (p(s(bool,'h4/const/bool/?!'(s(fun(A,bool),P)))) => p(s(bool,'h4/const/quotient/RES_EXISTS_EQUIV'(s(fun(A,fun(A,bool)),E),s(fun(A,bool),Q))))))).
