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;

theorem Th37:
  X is commutative BCK-algebra iff for I being Ideal of X holds I
  is commutative Ideal of X
proof
  thus X is commutative BCK-algebra implies for I being Ideal of X holds I is
  commutative Ideal of X
  proof
    assume X is commutative BCK-algebra;
    then {0.X} is commutative Ideal of X by Th36;
    hence thesis by Th35;
  end;
  assume for I being Ideal of X holds I is commutative Ideal of X;
  then {0.X} is commutative Ideal of X by Th35;
  hence thesis by Th36;
end;
