Power of Group Element in Kernel of Homomorphism iff Power of Image is Identity

Theorem
Let $G$ be a group whose identity is $e_G$.

Let $H$ be a group whose identity is $e_H$.

Let $\phi: G \to H$ be a (group) homomorphism.

Let $x^n \in \map \ker \phi$ for some integer $n$.

Then:
 * $\paren {\map \phi x}^n = e_H$