reserve Y for non empty set,
  a,b,c,d for Function of Y,BOOLEAN;

theorem
  a 'imp' b = a 'eqv' (a '&' b)
proof
  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 thesis by BVFUNC_4:8;
end;
