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

proof

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

end;

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