Lifting The Exponent Lemma for p=2

Theorem
Let $x, y \in \Z$ be odd integers.

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let:
 * $4 \mathrel \backslash x - y$

where $\backslash$ and $\nmid$ denote divisibility and non-divisibility respectively.

Then
 * $\nu_2 \left({x^n - y^n}\right) = \nu_2 \left({x - y}\right) + \nu_2 \left({n}\right)$

where $\nu_2$ denotes $2$-adic valuation.