let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds (a 'imp' (b '&' ('not' b))) 'imp' ('not' a) = I_el Y
let a, b be Function of Y,BOOLEAN; :: thesis: (a 'imp' (b '&' ('not' b))) 'imp' ('not' a) = I_el Y
for x being Element of Y holds ((a 'imp' (b '&' ('not' b))) 'imp' ('not' a)) . x = TRUE

proof

let x be Element of Y; :: thesis: ((a 'imp' (b '&' ('not' b))) 'imp' ('not' a)) . x = TRUE
((a 'imp' (b '&' ('not' b))) 'imp' ('not' a)) . x = ('not' ((a 'imp' (b '&' ('not' b))) . x)) 'or' (('not' a) . x) by BVFUNC_1:def 8
.= ('not' (('not' (a . x)) 'or' ((b '&' ('not' b)) . x))) 'or' (('not' a) . x) by BVFUNC_1:def 8
.= ((a . x) '&' (('not' (b . x)) 'or' ('not' (('not' b) . x)))) 'or' (('not' a) . x) by MARGREL1:def 20
.= ((a . x) '&' (('not' (b . x)) 'or' ('not' ('not' (b . x))))) 'or' (('not' a) . x) by MARGREL1:def 19
.= ((a . x) '&' TRUE) 'or' (('not' a) . x) by XBOOLEAN:102
.= (a . x) 'or' ('not' (a . x)) by MARGREL1:def 19
.= TRUE by XBOOLEAN:102 ;
hence ((a 'imp' (b '&' ('not' b))) 'imp' ('not' a)) . x = TRUE ; :: thesis: verum

end;

hence (a 'imp' (b '&' ('not' b))) 'imp' ('not' a) = I_el Y by BVFUNC_1:def 11; :: thesis: verum