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

proof

let x be Element of Y; :: thesis: ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = b . x
A5: ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = ((a 'imp' b) . x) '&' ((('not' a) 'imp' b) . x) by MARGREL1:def 21
.= (('not' (a . x)) 'or' (b . x)) '&' ((('not' a) 'imp' b) . x) by BVFUNC_1:def 11
.= (('not' (a . x)) 'or' (b . x)) '&' (('not' (('not' a) . x)) 'or' (b . x)) by BVFUNC_1:def 11
.= (('not' (a . x)) 'or' (b . x)) '&' ((a . x) 'or' (b . x)) by MARGREL1:def 20 ;

now

per cases ( a . x = TRUE or a . x = FALSE ) by XBOOLEAN:def 3;

case a . x = TRUE ; :: thesis: ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = b . x

then ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = (FALSE 'or' (b . x)) '&' (TRUE 'or' (b . x)) by A5, MARGREL1:41
.= (FALSE 'or' (b . x)) '&' TRUE by BINARITH:19
.= TRUE '&' (b . x) by BINARITH:7
.= b . x by MARGREL1:50 ;
hence ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = b . x ; :: thesis: verum

end;

case a . x = FALSE ; :: thesis: ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = b . x

then ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = (TRUE 'or' (b . x)) '&' (FALSE 'or' (b . x)) by A5, MARGREL1:41
.= TRUE '&' (FALSE 'or' (b . x)) by BINARITH:19
.= TRUE '&' (b . x) by BINARITH:7
.= b . x by MARGREL1:50 ;
hence ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = b . x ; :: thesis: verum

end;

end;

end;

hence ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = 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' a) 'imp' b) = b by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum