reserve Y for non empty set;

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