reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN holds (a 'imp' (b 'imp' c)
  ) = I_el(Y) & a 'imp' b = I_el(Y) & a = I_el(Y) implies c = I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
  assume that
A1: (a 'imp' (b 'imp' c)) = I_el(Y) and
A2: a 'imp' b = I_el(Y) and
A3: a = I_el(Y);
  for x being Element of Y holds c.x=TRUE
  proof
    let x be Element of Y;
    (a 'imp' (b 'imp' c)).x=TRUE by A1,BVFUNC_1:def 11;
    then 'not' a.x 'or' (b 'imp' c).x=TRUE by BVFUNC_1:def 8;
    then
A4: 'not' a.x 'or' ('not' b.x 'or' c.x)=TRUE by BVFUNC_1:def 8;
    (a 'imp' b).x=TRUE by A2,BVFUNC_1:def 11;
    then 'not' a.x 'or' b.x=TRUE by BVFUNC_1:def 8;
    then FALSE 'or' b.x=TRUE by A3,BVFUNC_1:def 11,MARGREL1:11;
    then b.x=TRUE by BINARITH:3;
    then FALSE 'or' ('not' TRUE 'or' c.x)=TRUE
          by A4,A3,BVFUNC_1:def 11,MARGREL1:11;
    then FALSE 'or' (FALSE 'or' c.x)=TRUE by MARGREL1:11;
    then (FALSE 'or' c.x)=TRUE by BINARITH:3;
    hence thesis by BINARITH:3;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
