let Y be non empty set ; :: thesis: for a, b being Element of Funcs (Y,BOOLEAN) holds (a 'imp' b) '&' ('not' b) = ('not' a) '&' ('not' b)
let a, b be Element of Funcs (Y,BOOLEAN); :: thesis: (a 'imp' b) '&' ('not' b) = ('not' a) '&' ('not' b)
consider k3 being Function such that
A1: (a 'imp' b) '&' ('not' b) = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: ('not' a) '&' ('not' b) = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds ((a 'imp' b) '&' ('not' b)) . x = (('not' a) '&' ('not' b)) . x

proof

let x be Element of Y; :: thesis: ((a 'imp' b) '&' ('not' b)) . x = (('not' a) '&' ('not' b)) . x
((a 'imp' b) '&' ('not' b)) . x = ((a 'imp' b) . x) '&' (('not' b) . x) by MARGREL1:def 20
.= (('not' b) . x) '&' (('not' (a . x)) 'or' (b . x)) by BVFUNC_1:def 8
.= ((('not' b) . x) '&' ('not' (a . x))) 'or' ((('not' b) . x) '&' (b . x)) by XBOOLEAN:8
.= ((('not' b) . x) '&' ('not' (a . x))) 'or' ((b . x) '&' ('not' (b . x))) by MARGREL1:def 19
.= ((('not' b) . x) '&' ('not' (a . x))) 'or' FALSE by XBOOLEAN:138
.= (('not' b) . x) '&' ('not' (a . x)) by BINARITH:3
.= (('not' b) . x) '&' (('not' a) . x) by MARGREL1:def 19
.= (('not' a) '&' ('not' b)) . x by MARGREL1:def 20 ;
hence ((a 'imp' b) '&' ('not' b)) . x = (('not' a) '&' ('not' b)) . x ; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence (a 'imp' b) '&' ('not' b) = ('not' a) '&' ('not' b) by A1, A2, A3, A4, FUNCT_1:2; :: thesis: verum