%   ORIGINAL: 'h4/thm/bool/LET_RAND_'
% 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 N M. P ('h4/const/bool/LET' (\x. N x) M) <=> 'h4/const/bool/LET' (\x. P (N x)) M
%   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. (!P N x. happ (happ (happ _1 P) N) x <=> happ P (happ N x)) ==> (!_0. (!N x. happ (happ _0 N) x = happ N x) ==> (!P N M. happ P ('h4/const/bool/LET' (happ _0 N) M) <=> 'h4/const/bool/LET' (happ (happ _1 P) N) M))
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_3f34228,V_3f34224]: ![F, G]: (![X]: s(V_3f34224,happ(s(fun(V_3f34228,V_3f34224),F),s(V_3f34228,X))) = s(V_3f34224,happ(s(fun(V_3f34228,V_3f34224),G),s(V_3f34228,X))) => s(fun(V_3f34228,V_3f34224),F) = s(fun(V_3f34228,V_3f34224),G))).
fof('h4/thm/bool/LET_RAND_', conjecture, ![B,A]: ![V__1]: (![P, N, X]: s(bool,happ(s(fun(A,bool),happ(s(fun(fun(A,B),fun(A,bool)),happ(s(fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),V__1),s(fun(B,bool),P))),s(fun(A,B),N))),s(A,X))) = s(bool,happ(s(fun(B,bool),P),s(B,happ(s(fun(A,B),N),s(A,X))))) => ![V__0]: (![N, X]: s(B,happ(s(fun(A,B),happ(s(fun(fun(A,B),fun(A,B)),V__0),s(fun(A,B),N))),s(A,X))) = s(B,happ(s(fun(A,B),N),s(A,X))) => ![P, N, M]: s(bool,happ(s(fun(B,bool),P),s(B,'h4/const/bool/LET'(s(fun(A,B),happ(s(fun(fun(A,B),fun(A,B)),V__0),s(fun(A,B),N))),s(A,M))))) = s(bool,'h4/const/bool/LET'(s(fun(A,bool),happ(s(fun(fun(A,B),fun(A,bool)),happ(s(fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),V__1),s(fun(B,bool),P))),s(fun(A,B),N))),s(A,M)))))).
