%   ORIGINAL: 'h4/thm/pred_set/DELETE_applied_'
% 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: !s x y. 'h4/const/pred_set/DELETE' s y x <=> 'h4/const/bool/IN' x s /\ ~(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: !s x y. 'h4/const/pred_set/DELETE' s y x <=> 'h4/const/bool/IN' x s /\ ~(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_3f17144,V_3f17140]: ![F, G]: (![X]: s(V_3f17140,happ(s(fun(V_3f17144,V_3f17140),F),s(V_3f17144,X))) = s(V_3f17140,happ(s(fun(V_3f17144,V_3f17140),G),s(V_3f17144,X))) => s(fun(V_3f17144,V_3f17140),F) = s(fun(V_3f17144,V_3f17140),G))).
fof('h4/thm/pred_set/DELETE_applied_', conjecture, ![A]: ![S, X, Y]: (p(s(bool,'h4/const/pred_set/DELETE'(s(fun(A,bool),S),s(A,Y),s(A,X)))) <=> (p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),S)))) & ~ (s(A,X) = s(A,Y))))).
