Complex Number equals Negative of Conjugate iff Wholly Imaginary

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

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

Then $\overline z = -z$ $z$ is wholly imaginary.

Proof
Let $z = x + i y$.

Then:

Hence by definition, $z$ is wholly imaginary.

Now suppose $z$ is wholly imaginary.

Then: