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

proof

let x be Element of Y; :: thesis: (a 'eqv' b) . x = ((a 'imp' b) '&' (b 'imp' a)) . x
(a 'eqv' b) . x = 'not' ((a . x) 'xor' (b . x)) by BVFUNC_1:def 9
.= (((a . x) 'or' ('not' (b . x))) '&' ('not' (a . x))) 'or' (((a . x) 'or' ('not' (b . x))) '&' (b . x)) by XBOOLEAN:8
.= ((('not' (a . x)) '&' (a . x)) 'or' (('not' (a . x)) '&' ('not' (b . x)))) 'or' ((b . x) '&' ((a . x) 'or' ('not' (b . x)))) by XBOOLEAN:8
.= ((('not' (a . x)) '&' (a . x)) 'or' (('not' (a . x)) '&' ('not' (b . x)))) 'or' (((b . x) '&' (a . x)) 'or' ((b . x) '&' ('not' (b . x)))) by XBOOLEAN:8
.= (FALSE 'or' (('not' (a . x)) '&' ('not' (b . x)))) 'or' (((b . x) '&' (a . x)) 'or' ((b . x) '&' ('not' (b . x)))) by XBOOLEAN:138
.= (('not' (a . x)) '&' ('not' (b . x))) 'or' (((b . x) '&' (a . x)) 'or' FALSE) by XBOOLEAN:138
.= ((('not' (a . x)) '&' ('not' (b . x))) 'or' (b . x)) '&' ((('not' (a . x)) '&' ('not' (b . x))) 'or' (a . x)) by XBOOLEAN:9
.= (((b . x) 'or' ('not' (a . x))) '&' ((b . x) 'or' ('not' (b . x)))) '&' ((a . x) 'or' (('not' (a . x)) '&' ('not' (b . x)))) by XBOOLEAN:9
.= (((b . x) 'or' ('not' (a . x))) '&' ((b . x) 'or' ('not' (b . x)))) '&' (((a . x) 'or' ('not' (a . x))) '&' ((a . x) 'or' ('not' (b . x)))) by XBOOLEAN:9
.= (((b . x) 'or' ('not' (a . x))) '&' TRUE) '&' (((a . x) 'or' ('not' (a . x))) '&' ((a . x) 'or' ('not' (b . x)))) by XBOOLEAN:102
.= ((b . x) 'or' ('not' (a . x))) '&' (TRUE '&' ((a . x) 'or' ('not' (b . x)))) by XBOOLEAN:102
.= ((a 'imp' b) . x) '&' (('not' (b . x)) 'or' (a . x)) by BVFUNC_1:def 8
.= ((a 'imp' b) . x) '&' ((b 'imp' a) . x) by BVFUNC_1:def 8
.= ((a 'imp' b) '&' (b 'imp' a)) . x by MARGREL1:def 20 ;
hence (a 'eqv' b) . x = ((a 'imp' b) '&' (b 'imp' a)) . x ; :: thesis: verum

end;

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