let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs Y,BOOLEAN holds a 'imp' (b 'or' c) = (a '&' ('not' b)) 'imp' c
let a, b, c be Element of Funcs Y,BOOLEAN ; :: thesis: a 'imp' (b 'or' c) = (a '&' ('not' b)) 'imp' c
A1: for x being Element of Y holds (a 'imp' (b 'or' c)) . x = ((a '&' ('not' b)) 'imp' c) . x

proof

let x be Element of Y; :: thesis: (a 'imp' (b 'or' c)) . x = ((a '&' ('not' b)) 'imp' c) . x
((a '&' ('not' b)) 'imp' c) . x = ('not' ((a '&' ('not' b)) . x)) 'or' (c . x) by BVFUNC_1:def 11
.= (('not' (a . x)) 'or' ('not' (('not' b) . x))) 'or' (c . x) by MARGREL1:def 21
.= (('not' (a . x)) 'or' (b . x)) 'or' (c . x) by MARGREL1:def 20
.= ('not' (a . x)) 'or' ((b . x) 'or' (c . x)) by BINARITH:20
.= ('not' (a . x)) 'or' ((b 'or' c) . x) by BVFUNC_1:def 7
.= (a 'imp' (b 'or' c)) . x by BVFUNC_1:def 11 ;
hence (a 'imp' (b 'or' c)) . x = ((a '&' ('not' b)) 'imp' c) . x ; :: thesis: verum

end;

consider k3 being Function such that
A2: ( a 'imp' (b 'or' c) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A3: ( (a '&' ('not' b)) 'imp' c = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
( Y = dom k3 & Y = dom k4 & ( for u being set st u in Y holds
k3 . u = k4 . u ) ) by A1, A2, A3;
hence a 'imp' (b 'or' c) = (a '&' ('not' b)) 'imp' c by A2, A3, FUNCT_1:9; :: thesis: verum