%   ORIGINAL: 'h4/thm/pred_set/INJ_IFF_'
% 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: !t s f. 'h4/const/pred_set/INJ' f s t <=> (!x. 'h4/const/bool/IN' x s ==> 'h4/const/bool/IN' (f x) t) /\ (!x y. 'h4/const/bool/IN' x s /\ 'h4/const/bool/IN' y s ==> (f x = f 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: !t s f. 'h4/const/pred_set/INJ' f s t <=> (!x. 'h4/const/bool/IN' x s ==> 'h4/const/bool/IN' (happ f x) t) /\ (!x y. 'h4/const/bool/IN' x s /\ 'h4/const/bool/IN' y s ==> (happ f x = happ f 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_3f18420,V_3f18416]: ![F, G]: (![X]: s(V_3f18416,happ(s(fun(V_3f18420,V_3f18416),F),s(V_3f18420,X))) = s(V_3f18416,happ(s(fun(V_3f18420,V_3f18416),G),s(V_3f18420,X))) => s(fun(V_3f18420,V_3f18416),F) = s(fun(V_3f18420,V_3f18416),G))).
fof('h4/thm/pred_set/INJ_IFF_', conjecture, ![B,A]: ![T, S, F]: (p(s(bool,'h4/const/pred_set/INJ'(s(fun(A,B),F),s(fun(A,bool),S),s(fun(B,bool),T)))) <=> (![X]: (p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),S)))) => p(s(bool,'h4/const/bool/IN'(s(B,happ(s(fun(A,B),F),s(A,X))),s(fun(B,bool),T))))) & ![X, Y]: ((p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),S)))) & p(s(bool,'h4/const/bool/IN'(s(A,Y),s(fun(A,bool),S))))) => (s(B,happ(s(fun(A,B),F),s(A,X))) = s(B,happ(s(fun(A,B),F),s(A,Y))) <=> s(A,X) = s(A,Y)))))).
