\begin{definition}[Definition:Semantic Equivalence/Boolean Interpretations/Definition 1]
Let $\mathbf A, \mathbf B$ be WFFs of propositional logic.
Then $\mathbf A$ and $\mathbf B$ are '''equivalent for boolean interpretations''' {{iff}}:
:$\mathbf A \models_{\mathrm{BI}} \mathbf B$ 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''' {{iff}}:
:$\map v {\mathbf A} = T$ {{iff}} $\map v {\mathbf B} = T$
for all boolean interpretations $v$.
\end{definition}