Definition:Relative Semantic Equivalence/WFF

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Let $\mathcal F$ be a theory in the language of predicate logic.

Let $\mathbf A, \mathbf B$ be WFFs.

Let $\mathbf C$ be the universal closure of $\mathbf A \iff \mathbf B$.

Then $\mathbf A$ and $\mathbf B$ are semantically equivalent with respect to $\mathcal F$ if and only if:

$\mathcal F \models_{\mathrm{PL}} \mathbf C$

That is, iff $\mathbf C$ is a semantic consequence of $\mathcal F$.

Also see