reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN st (a 'imp' b)=I_el(Y) & (
  b 'imp' c)=I_el(Y) holds (a 'imp' c)=I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
  assume that
A1: (a 'imp' b)=I_el(Y) and
A2: (b 'imp' c)=I_el(Y);
  for x being Element of Y holds (a 'imp' c).x=TRUE
  proof
    let x be Element of Y;
A3: ('not' b.x)=TRUE or ('not' b.x)=FALSE by XBOOLEAN:def 3;
A4: (a 'imp' c).x=('not' a.x) 'or' c.x by BVFUNC_1:def 8;
    (b 'imp' c).x=TRUE by A2,BVFUNC_1:def 11;
    then
A5: ('not' b.x) 'or' c.x=TRUE by BVFUNC_1:def 8;
    (a 'imp' b).x=TRUE by A1,BVFUNC_1:def 11;
    then
A6: ('not' a.x) 'or' b.x=TRUE by BVFUNC_1:def 8;
A7: ('not' a.x)=TRUE or ('not' a.x)=FALSE by XBOOLEAN:def 3;
    now
      per cases by A6,A7,A5,A3,BINARITH:3;
      case
        ('not' a.x)=TRUE & ('not' b.x)=TRUE;
        hence thesis by A4,BINARITH:10;
      end;
      case
        ('not' a.x)=TRUE & c.x=TRUE;
        hence thesis by A4;
      end;
      case
        b.x=TRUE & ('not' b.x)=TRUE;
        hence thesis by MARGREL1:11;
      end;
      case
        b.x=TRUE & c.x=TRUE;
        hence thesis by A4,BINARITH:10;
      end;
    end;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
