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

proof

let x be Element of Y; :: thesis: (('not' (a 'or' b)) 'imp' (('not' a) '&' ('not' b))) . x = TRUE
(('not' (a 'or' b)) 'imp' (('not' a) '&' ('not' b))) . x = ('not' (('not' (a 'or' b)) . x)) 'or' ((('not' a) '&' ('not' b)) . x) by BVFUNC_1:def 8
.= ((a 'or' b) . x) 'or' ((('not' a) '&' ('not' b)) . x) by MARGREL1:def 19
.= ((a . x) 'or' (b . x)) 'or' ((('not' a) '&' ('not' b)) . x) by BVFUNC_1:def 4
.= ((a . x) 'or' (b . x)) 'or' ((('not' a) . x) '&' (('not' b) . x)) by MARGREL1:def 20
.= ((a . x) 'or' (b . x)) 'or' (('not' (a . x)) '&' (('not' b) . x)) by MARGREL1:def 19
.= ((a . x) 'or' (b . x)) 'or' (('not' (a . x)) '&' ('not' (b . x))) by MARGREL1:def 19
.= (((a . x) 'or' (b . x)) 'or' ('not' (a . x))) '&' (((a . x) 'or' (b . x)) 'or' ('not' (b . x))) by XBOOLEAN:9
.= (((a . x) 'or' ('not' (a . x))) 'or' (b . x)) '&' ((a . x) 'or' ((b . x) 'or' ('not' (b . x))))
.= (TRUE 'or' (b . x)) '&' ((a . x) 'or' ((b . x) 'or' ('not' (b . x)))) by XBOOLEAN:102
.= TRUE '&' ((a . x) 'or' TRUE) by XBOOLEAN:102
.= TRUE ;
hence (('not' (a 'or' b)) 'imp' (('not' a) '&' ('not' b))) . x = TRUE ; :: thesis: verum

end;

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