let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds a 'eqv' b = (a 'imp' b) '&' (b 'imp' a)
let a, b be Function of Y,BOOLEAN; :: thesis: a 'eqv' b = (a 'imp' b) '&' (b 'imp' a)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: (a 'eqv' b) . x = ((a 'imp' b) '&' (b 'imp' a)) . x
thus (a 'eqv' b) . x = (a . x) <=> (b . x) by BVFUNC_1:def 9
.= ((a . x) => (b . x)) '&' ((b . x) => (a . x))
.= ((a 'imp' b) . x) '&' ((b . x) => (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 ; :: thesis: verum