let Y be non empty set ; :: thesis: for a, b, c, d being Element of Funcs (Y,BOOLEAN) holds ((a '&' b) '&' c) 'imp' d = ((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)
let a, b, c, d be Element of Funcs (Y,BOOLEAN); :: thesis: ((a '&' b) '&' c) 'imp' d = ((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)
consider k3 being Function such that
A1: ((a '&' b) '&' c) 'imp' d = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: ((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d) = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds (((a '&' b) '&' c) 'imp' d) . x = (((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)) . x

proof

let x be Element of Y; :: thesis: (((a '&' b) '&' c) 'imp' d) . x = (((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)) . x
(((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)) . x = (((a 'imp' d) 'or' (b 'imp' d)) . x) 'or' ((c 'imp' d) . x) by BVFUNC_1:def 4
.= (((a 'imp' d) . x) 'or' ((b 'imp' d) . x)) 'or' ((c 'imp' d) . x) by BVFUNC_1:def 4
.= ((('not' (a . x)) 'or' (d . x)) 'or' ((b 'imp' d) . x)) 'or' ((c 'imp' d) . x) by BVFUNC_1:def 8
.= ((('not' (a . x)) 'or' (d . x)) 'or' (('not' (b . x)) 'or' (d . x))) 'or' ((c 'imp' d) . x) by BVFUNC_1:def 8
.= ((('not' (a . x)) 'or' (d . x)) 'or' (('not' (b . x)) 'or' (d . x))) 'or' (('not' (c . x)) 'or' (d . x)) by BVFUNC_1:def 8
.= ((('not' (a . x)) 'or' ((d . x) 'or' ('not' (b . x)))) 'or' (d . x)) 'or' (('not' (c . x)) 'or' (d . x)) by BINARITH:11
.= (((('not' (a . x)) 'or' ('not' (b . x))) 'or' (d . x)) 'or' (d . x)) 'or' (('not' (c . x)) 'or' (d . x)) by BINARITH:11
.= ((('not' (a . x)) 'or' ('not' (b . x))) 'or' ((d . x) 'or' (d . x))) 'or' (('not' (c . x)) 'or' (d . x)) by BINARITH:11
.= (('not' ((a . x) '&' (b . x))) 'or' ((d . x) 'or' ('not' (c . x)))) 'or' (d . x) by BINARITH:11
.= ((('not' ((a . x) '&' (b . x))) 'or' ('not' (c . x))) 'or' (d . x)) 'or' (d . x) by BINARITH:11
.= (('not' ((a . x) '&' (b . x))) 'or' ('not' (c . x))) 'or' ((d . x) 'or' (d . x)) by BINARITH:11
.= ('not' (((a '&' b) . x) '&' (c . x))) 'or' (d . x) by MARGREL1:def 20
.= ('not' (((a '&' b) '&' c) . x)) 'or' (d . x) by MARGREL1:def 20
.= (((a '&' b) '&' c) 'imp' d) . x by BVFUNC_1:def 8 ;
hence (((a '&' b) '&' c) 'imp' d) . x = (((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)) . x ; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence ((a '&' b) '&' c) 'imp' d = ((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d) by A1, A2, A3, A4, FUNCT_1:2; :: thesis: verum