let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN st a 'imp' b = I_el Y holds
(a '&' c) 'imp' (b '&' c) = I_el Y
let a, b, c be Function of Y,BOOLEAN; :: thesis: ( a 'imp' b = I_el Y implies (a '&' c) 'imp' (b '&' c) = I_el Y )
assume A1: a 'imp' b = I_el Y ; :: thesis: (a '&' c) 'imp' (b '&' c) = I_el Y
for x being Element of Y holds ((a '&' c) 'imp' (b '&' c)) . x = TRUE

proof

let x be Element of Y; :: thesis: ((a '&' c) 'imp' (b '&' c)) . x = TRUE
(a 'imp' b) . x = TRUE by A1, BVFUNC_1:def 11;
then A2: ('not' (a . x)) 'or' (b . x) = TRUE by BVFUNC_1:def 8;
((a '&' c) 'imp' (b '&' c)) . x = ('not' ((a '&' c) . x)) 'or' ((b '&' c) . x) by BVFUNC_1:def 8
.= ('not' ((a . x) '&' (c . x))) 'or' ((b '&' c) . x) by MARGREL1:def 20
.= (('not' (a . x)) 'or' ('not' (c . x))) 'or' ((b . x) '&' (c . x)) by MARGREL1:def 20
.= ((('not' (c . x)) 'or' ('not' (a . x))) 'or' (b . x)) '&' ((('not' (a . x)) 'or' ('not' (c . x))) 'or' (c . x)) by XBOOLEAN:9
.= (('not' (c . x)) 'or' (('not' (a . x)) 'or' (b . x))) '&' (('not' (a . x)) 'or' (('not' (c . x)) 'or' (c . x)))
.= (('not' (c . x)) 'or' (('not' (a . x)) 'or' (b . x))) '&' (('not' (a . x)) 'or' TRUE) by XBOOLEAN:102
.= TRUE by A2 ;
hence ((a '&' c) 'imp' (b '&' c)) . x = TRUE ; :: thesis: verum

end;

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