let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds a 'imp' (b 'imp' (a '&' b)) = I_el Y
let a, b be Function of Y,BOOLEAN; :: thesis: a 'imp' (b 'imp' (a '&' b)) = I_el Y
for x being Element of Y holds (a 'imp' (b 'imp' (a '&' b))) . x = TRUE

proof

let x be Element of Y; :: thesis: (a 'imp' (b 'imp' (a '&' b))) . x = TRUE
(a 'imp' (b 'imp' (a '&' b))) . x = ('not' (a . x)) 'or' ((b 'imp' (a '&' b)) . x) by BVFUNC_1:def 8
.= ('not' (a . x)) 'or' (('not' (b . x)) 'or' ((a '&' b) . x)) by BVFUNC_1:def 8
.= ('not' (a . x)) 'or' (('not' (b . x)) 'or' ((a . x) '&' (b . x))) by MARGREL1:def 20
.= ('not' (a . x)) 'or' ((('not' (b . x)) 'or' (a . x)) '&' (('not' (b . x)) 'or' (b . x))) by XBOOLEAN:9
.= ('not' (a . x)) 'or' (TRUE '&' (('not' (b . x)) 'or' (a . x))) by XBOOLEAN:102
.= (('not' (a . x)) 'or' (a . x)) 'or' ('not' (b . x))
.= TRUE 'or' ('not' (b . x)) by XBOOLEAN:102
.= TRUE ;
hence (a 'imp' (b 'imp' (a '&' b))) . x = TRUE ; :: thesis: verum

end;

hence a 'imp' (b 'imp' (a '&' b)) = I_el Y by BVFUNC_1:def 11; :: thesis: verum