reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN holds ((a '&' b) 'imp' c)
  'imp' (a 'imp' (b 'imp' c))=I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
  for x being Element of Y holds (((a '&' b) 'imp' c) 'imp' (a 'imp' (b
  'imp' c))).x=TRUE
  proof
    let x be Element of Y;
    (((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c))).x ='not' ((a '&' b)
    'imp' c).x 'or' (a 'imp' (b 'imp' c)).x by BVFUNC_1:def 8
      .='not'( 'not' (a '&' b).x 'or' c.x) 'or' (a 'imp' (b 'imp' c)).x by
BVFUNC_1:def 8
      .='not'( 'not' (a.x '&' b.x) 'or' c.x) 'or' (a 'imp' (b 'imp' c)
    ).x by MARGREL1:def 20
      .='not'( 'not'( a.x '&' b.x) 'or' c.x) 'or' ('not' a.x 'or' (b
    'imp' c).x) by BVFUNC_1:def 8
      .='not'( ('not' a.x 'or' 'not' b.x) 'or' c.x) 'or' ('not' a.x
    'or' ('not' b.x 'or' c.x)) by BVFUNC_1:def 8
      .=TRUE by XBOOLEAN:102;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
