%   ORIGINAL: 'h4/thm/ind_type/ISO_USAGE_'
% 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: !g f. 'h4/const/ind_type/ISO' f g ==> (!P. (!x. P x) <=> (!x. P (g x))) /\ (!P. (?x. P x) <=> (?x. P (g x))) /\ (!a b. a = g b <=> f a = b)
%   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: !g f. 'h4/const/ind_type/ISO' f g ==> (!P. (!x. happ P x) <=> (!x. happ P (happ g x))) /\ (!P. (?x. happ P x) <=> (?x. happ P (happ g x))) /\ (!a b. a = happ g b <=> happ f a = b)
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_3f74365,V_3f74361]: ![F, G]: (![X]: s(V_3f74361,happ(s(fun(V_3f74365,V_3f74361),F),s(V_3f74365,X))) = s(V_3f74361,happ(s(fun(V_3f74365,V_3f74361),G),s(V_3f74365,X))) => s(fun(V_3f74365,V_3f74361),F) = s(fun(V_3f74365,V_3f74361),G))).
fof('h4/thm/ind_type/ISO_USAGE_', conjecture, ![A,B]: ![G, F]: (p(s(bool,'h4/const/ind_type/ISO'(s(fun(A,B),F),s(fun(B,A),G)))) => (![P]: (![X]: p(s(bool,happ(s(fun(A,bool),P),s(A,X)))) <=> ![X]: p(s(bool,happ(s(fun(A,bool),P),s(A,happ(s(fun(B,A),G),s(B,X))))))) & (![P]: (?[X]: p(s(bool,happ(s(fun(A,bool),P),s(A,X)))) <=> ?[X]: p(s(bool,happ(s(fun(A,bool),P),s(A,happ(s(fun(B,A),G),s(B,X))))))) & ![A0, B0]: (s(A,A0) = s(A,happ(s(fun(B,A),G),s(B,B0))) <=> s(B,happ(s(fun(A,B),F),s(A,A0))) = s(B,B0)))))).
