Equivalence of Definitions of Semantic Equivalence for Predicate Logic

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

Proof
Let $\AA$ be a structure for predicate logic.

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

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


 * $\map {f^\leftrightarrow} {\map {\operatorname{val}_\AA} {\mathbf A} \sqbrk \sigma, \map {\operatorname{val}_\AA} {\mathbf B} \sqbrk \sigma}$

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


 * $\map {\operatorname{val}_\AA} {\mathbf A} \sqbrk \sigma = \map {\operatorname{val}_\AA} {\mathbf B} \sqbrk \sigma$

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


 * $\map {\operatorname{val}_\AA} {\mathbf A} \sqbrk \sigma = \T$ $\map {\operatorname{val}_\AA} {\mathbf B} \sqbrk \sigma = \T$

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


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

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

The result follows by definition of tautology.