Equivalence of Definitions of Semantic Equivalence for Predicate Logic

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

The following definitions of semantic equivalence of $\mathbf A$ and $\mathbf B$ are equivalent:

Proof
Let $\mathcal A$ be a structure for predicate logic.

Let $\sigma$ be an assignment for $\mathbf A \iff \mathbf B$ in $\mathcal A$.

Then the value of $\mathbf A \iff \mathbf B$ under $\sigma$ is given by:


 * $f^\leftrightarrow \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right], \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma}\right] }\right)$

and from the definition of $f^\leftrightarrow$ we see that $\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf A \iff \mathbf B$ iff:


 * $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right] = \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma}\right]$

Because the possible values are just $T$ and $F$, this is equivalent to:


 * $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right] = T$ iff $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma}\right] = T$

which by definition of $\mathrm{PL_A}$-model amounts to:


 * $\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf A$ iff $\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf B$

Because $\mathcal A$ and $\sigma$ were arbitrary, the above equivalence holds for all such $\mathcal A$ and $\sigma$.

The result follows by definition of tautology.