Equation of Imaginary Axis in Complex Plane

Theorem
Let $\C$ be the complex plane.

Let $z \in \C$ be subject to the condition:
 * $\cmod {z - 1} = \cmod {z + 1}$

where $\cmod {\, \cdot \,}$ denotes complex modulus.

Then the locus of $z$ is the imaginary axis.

Proof
The result follows by definition of imaginary axis.