let Y be non empty set ; :: thesis: for a, b, c, d being Function of Y,BOOLEAN st a 'imp' b = I_el Y & c 'imp' d = I_el Y holds
(a '&' c) 'imp' (b '&' d) = I_el Y
let a, b, c, d be Function of Y,BOOLEAN; :: thesis: ( a 'imp' b = I_el Y & c 'imp' d = I_el Y implies (a '&' c) 'imp' (b '&' d) = I_el Y )
assume that
A1: a 'imp' b = I_el Y and
A2: c 'imp' d = I_el Y ; :: thesis: (a '&' c) 'imp' (b '&' d) = I_el Y
for x being Element of Y holds ((a '&' c) 'imp' (b '&' d)) . x = TRUE

proof

let x be Element of Y; :: thesis: ((a '&' c) 'imp' (b '&' d)) . x = TRUE
(a 'imp' b) . x = TRUE by A1, BVFUNC_1:def 11;
then A3: ('not' (a . x)) 'or' (b . x) = TRUE by BVFUNC_1:def 8;
(c 'imp' d) . x = TRUE by A2, BVFUNC_1:def 11;
then A4: ('not' (c . x)) 'or' (d . x) = TRUE by BVFUNC_1:def 8;
((a '&' c) 'imp' (b '&' d)) . x = ('not' ((a '&' c) . x)) 'or' ((b '&' d) . x) by BVFUNC_1:def 8
.= ('not' ((a . x) '&' (c . x))) 'or' ((b '&' d) . x) by MARGREL1:def 20
.= (('not' (a . x)) 'or' ('not' (c . x))) 'or' ((b . x) '&' (d . x)) by MARGREL1:def 20
.= ((('not' (c . x)) 'or' ('not' (a . x))) 'or' (b . x)) '&' ((('not' (a . x)) 'or' ('not' (c . x))) 'or' (d . x)) by XBOOLEAN:9
.= (('not' (c . x)) 'or' (('not' (a . x)) 'or' (b . x))) '&' ((('not' (a . x)) 'or' ('not' (c . x))) 'or' (d . x))
.= TRUE '&' (('not' (a . x)) 'or' TRUE) by A3, A4, BINARITH:11
.= TRUE ;
hence ((a '&' c) 'imp' (b '&' d)) . x = TRUE ; :: thesis: verum

end;

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