%   ORIGINAL: 'h4/thm/res_quan/RES_FORALL_FORALL_'
% 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 R x. (!x0. 'h4/const/bool/RES_FORALL' P (\i. R i x0)) <=> 'h4/const/bool/RES_FORALL' P (\i. !x0. R i x0)
%   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: !_1. (!R i. happ (happ _1 R) i <=> (!x0. happ (happ R i) x0)) ==> (!_0. (!R x0 i. happ (happ (happ _0 R) x0) i <=> happ (happ R i) x0) ==> (!P R x. (!x0. 'h4/const/bool/RES_FORALL' P (happ (happ _0 R) x0)) <=> 'h4/const/bool/RES_FORALL' P (happ _1 R)))
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_3f30853,V_3f30849]: ![F, G]: (![X]: s(V_3f30849,happ(s(fun(V_3f30853,V_3f30849),F),s(V_3f30853,X))) = s(V_3f30849,happ(s(fun(V_3f30853,V_3f30849),G),s(V_3f30853,X))) => s(fun(V_3f30853,V_3f30849),F) = s(fun(V_3f30853,V_3f30849),G))).
fof('h4/thm/res_quan/RES_FORALL_FORALL_', conjecture, ![A,B]: ![V__1]: (![R, I]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(fun(A,fun(B,bool)),fun(A,bool)),V__1),s(fun(A,fun(B,bool)),R))),s(A,I)))) <=> ![X0]: p(s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),R),s(A,I))),s(B,X0))))) => ![V__0]: (![R, X0, I]: s(bool,happ(s(fun(A,bool),happ(s(fun(B,fun(A,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),V__0),s(fun(A,fun(B,bool)),R))),s(B,X0))),s(A,I))) = s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),R),s(A,I))),s(B,X0))) => ![P, R, X]: (![X0]: p(s(bool,'h4/const/bool/RES_FORALL'(s(fun(A,bool),P),s(fun(A,bool),happ(s(fun(B,fun(A,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),V__0),s(fun(A,fun(B,bool)),R))),s(B,X0)))))) <=> p(s(bool,'h4/const/bool/RES_FORALL'(s(fun(A,bool),P),s(fun(A,bool),happ(s(fun(fun(A,fun(B,bool)),fun(A,bool)),V__1),s(fun(A,fun(B,bool)),R)))))))))).
