reserve X for BCI-algebra;
reserve X1 for non empty Subset of X;
reserve A,I for Ideal of X;
reserve x,y,z for Element of X;
reserve a for Element of A;
reserve X for BCK-algebra;
reserve X for BCI-algebra;
reserve X for BCK-algebra;
reserve I for Ideal of X;
reserve I for Ideal of X;
reserve X for BCK-algebra;
reserve I for Ideal of X;

theorem Th50:
  I is implicative-ideal of X iff for x,y being Element of X st x\
  (y\x) in I holds x in I
proof
A1: (for x,y being Element of X st x\(y\x) in I holds x in I) implies I is
  implicative-ideal of X
  proof
    assume
A2: for x,y being Element of X st x\(y\x) in I holds x in I;
A3: for x,y,z being Element of X st (x\(y\x))\z in I & z in I holds x in I
    proof
      let x,y,z be Element of X;
      assume (x\(y\x))\z in I & z in I;
      then x\(y\x) in I by BCIALG_1:def 18;
      hence thesis by A2;
    end;
    0.X in I by BCIALG_1:def 18;
    hence thesis by A3,Def7;
  end;
  I is implicative-ideal of X implies for x,y being Element of X st x\(y\x
  ) in I holds x in I
  proof
    assume
A4: I is implicative-ideal of X;
    let x,y be Element of X;
    assume x\(y\x) in I;
    then
A5: x\(y\x)\0.X in I by BCIALG_1:2;
    thus thesis by A4,A5,Def7;
  end;
  hence thesis by A1;
end;
