Can someone please tell me where I can find a citeable reference for the following result:

Call the metric space $(X, d)$ "locally complete" if for every $x \in X$ there a neighbourhood of $x$ which is complete under $d$.

If $(X,d)$ is locally complete and separable then there exists a metric $d'$ on $X$ such that $(X,d) \to (X,d')$ is a homeomorphism and $(X, d')$ is complete.

This result follows immediately from Alexandrov's theorem that a $G_\delta$ subset of a Polish space is Polish, but I'd rather find a statement in the literature of the above more straightforward result if there is one.