reserve Y for non empty set;

theorem Th42:
  for a,b being Function of Y,BOOLEAN holds (a 'eqv' b)
  'imp' (a 'imp' b)=I_el(Y)
proof
  let a,b be Function of Y,BOOLEAN;
  for x being Element of Y holds ((a 'eqv' b) 'imp' (a 'imp' b)).x=TRUE
  proof
    let x be Element of Y;
    ((a 'eqv' b) 'imp' (a 'imp' b)).x ='not' (a 'eqv' b).x 'or' (a 'imp' b
    ).x by BVFUNC_1:def 8
      .=(a.x 'xor' b.x) 'or' (a 'imp' b).x by BVFUNC_1:def 9
      .=(('not' a.x '&' b.x) 'or' (a.x '&' 'not' b.x)) 'or' 'not'( (
    a).x '&' 'not' b.x) by BVFUNC_1:def 8
      .=('not' a.x '&' b.x) 'or' ((a.x '&' 'not' b.x) 'or' 'not'( (a
    ).x '&' 'not' b.x))
      .=('not' a.x '&' b.x) 'or' TRUE by XBOOLEAN:102
      .=TRUE;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
