Complex Number equals Conjugate iff Wholly Real

Theorem
Let $z \in \C$ be a complex number.

Let $\overline {z}$ be the complex conjugate of $z$.

Then $z = \overline z$ iff $z$ is wholly real.

Proof
Let $z = x + i y$.

Then:

Hence by definition, $z$ is wholly real.

Now suppose $z$ is wholly real.

Then: