Satisfiability preserved in Supersignature

Theorem
Let $\LL, \LL'$ be signatures for the language of predicate logic.

Let $\LL'$ be a supersignature of $\LL$.

Let $\Sigma$ be a set of $\LL$-sentences.

Then the following are equivalent:


 * $\AA \models_{\mathrm{PL} } \Sigma$ for some $\LL$-structure $\AA$
 * $\AA' \models_{\mathrm{PL} } \Sigma$ for some $\LL'$-structure $\AA'$

where $\models_{\mathrm{PL} }$ is the models relation.

That is to say, the notion of satisfiability is preserved in passing to a supersignature.