%   ORIGINAL: 'h4/thm/list/NOT_EVERY_'
% 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 l. ~'h4/const/list/EVERY' P l <=> 'h4/const/list/EXISTS' ('h4/const/combin/o' $not P) l
%   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 l. ~'h4/const/list/EVERY' P l <=> 'h4/const/list/EXISTS' ('h4/const/combin/o' $not P) l
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f3913,V_3f3909]: ![F, G]: (![X]: s(V_3f3909,happ(s(fun(V_3f3913,V_3f3909),F),s(V_3f3913,X))) = s(V_3f3909,happ(s(fun(V_3f3913,V_3f3909),G),s(V_3f3913,X))) => s(fun(V_3f3913,V_3f3909),F) = s(fun(V_3f3913,V_3f3909),G))).
fof('h4/thm/list/NOT_EVERY_', conjecture, ![A]: ![P, L]: (~ (p(s(bool,'h4/const/list/EVERY'(s(fun(A,bool),P),s('h4/type/list/list'(A),L))))) <=> p(s(bool,'h4/const/list/EXISTS'(s(fun(A,bool),'h4/const/combin/o'(s(fun(bool,bool),'$not'),s(fun(A,bool),P))),s('h4/type/list/list'(A),L)))))).
