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

proof

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

end;

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