Semantic Consequence preserved in Supersignature

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

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

Let $\mathbf A$ be an $\LL$-sentence.

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

Then the following are equivalent:


 * $\AA \models_{\mathrm{PL} } \mathbf A$ for all $\LL$-structures $\AA$ for which $\AA \models_{\mathrm{PL} } \Sigma$
 * $\AA' \models_{\mathrm{PL} } \mathbf A$ for all $\LL'$-structures $\AA'$ for which $\AA' \models_{\mathrm{PL} } \Sigma$

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

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