let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds a 'imp' b = a 'eqv' (a '&' b)
let a, b be Function of Y,BOOLEAN; :: thesis: a 'imp' b = a 'eqv' (a '&' b)
a 'eqv' (a '&' b) = (a 'imp' (a '&' b)) '&' ((a '&' b) 'imp' a) by BVFUNC_4:7
.= (('not' a) 'or' (a '&' b)) '&' ((a '&' b) 'imp' a) by BVFUNC_4:8
.= (('not' a) 'or' (a '&' b)) '&' (('not' (a '&' b)) 'or' a) by BVFUNC_4:8
.= ((('not' a) 'or' a) '&' (('not' a) 'or' b)) '&' (('not' (a '&' b)) 'or' a) by BVFUNC_1:11
.= ((I_el Y) '&' (('not' a) 'or' b)) '&' (('not' (a '&' b)) 'or' a) by BVFUNC_4:6
.= (('not' a) 'or' b) '&' (('not' (a '&' b)) 'or' a) by BVFUNC_1:6
.= (('not' a) 'or' b) '&' ((('not' a) 'or' ('not' b)) 'or' a) by BVFUNC_1:14
.= (('not' a) 'or' b) '&' (('not' b) 'or' (('not' a) 'or' a)) by BVFUNC_1:8
.= (('not' a) 'or' b) '&' (('not' b) 'or' (I_el Y)) by BVFUNC_4:6
.= (('not' a) 'or' b) '&' (I_el Y) by BVFUNC_1:10
.= ('not' a) 'or' b by BVFUNC_1:6 ;
hence a 'imp' b = a 'eqv' (a '&' b) by BVFUNC_4:8; :: thesis: verum