Definition:Semantic Equivalence/Boolean Interpretations/Definition 1

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Let $\mathbf A, \mathbf B$ be WFFs of propositional logic.

Then $\mathbf A$ and $\mathbf B$ are equivalent for boolean interpretations if and only if:

$\mathbf A \models_{\mathrm{BI}} \mathbf B$ and $\mathbf B \models_{\mathrm{BI}} \mathbf A$

that is, each is a semantic consequence of the other.

That is to say, $\mathbf A$ and $\mathbf B$ are equivalent if and only if:

$\map v {\mathbf A} = T$ if and only if $\map v {\mathbf B} = T$

for all boolean interpretations $v$.

Also see