reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN holds c 'imp' (b 'imp' a)=
  I_el(Y) implies b 'imp' (c 'imp' a)=I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
  assume
A1: c 'imp' (b 'imp' a)=I_el(Y);
  for x being Element of Y holds (b 'imp' (c 'imp' a)).x=TRUE
  proof
    let x be Element of Y;
    (c 'imp' (b 'imp' a)).x=TRUE by A1,BVFUNC_1:def 11;
    then 'not' c.x 'or' (b 'imp' a).x=TRUE by BVFUNC_1:def 8;
    then
A2: 'not' c.x 'or' ('not' b.x 'or' a.x)=TRUE by BVFUNC_1:def 8;
    (b 'imp' (c 'imp' a)).x ='not' b.x 'or' (c 'imp' a).x by BVFUNC_1:def 8
      .='not' b.x 'or' ('not' c.x 'or' a.x) by BVFUNC_1:def 8
      .=TRUE by A2,BINARITH:11;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
