Substitution Theorem for Well-Formed Formulas/Corollary

Corollary to Substitution Theorem for Well-Formed Formulas
Let $\mathbf A \left({x_1, \ldots, x_n}\right)$ be a WFF whose free variables are among $x_1, \ldots, x_n$.

Let $\tau_1, \ldots, \tau_n$ be closed terms, and let each $\tau_i$ be freely substitutable for $x_i$.

Let $a_i = \operatorname{val}_{\mathcal A} \left({\tau_i}\right)$, where $\operatorname{val}_{\mathcal A}$ denotes the value of $\tau$ in $\mathcal A$.

Then:


 * $\mathcal A \models_{\mathrm{PL}} \mathbf A \left({\tau_1, \ldots, \tau_n}\right)$ iff $\mathcal A \models_{\mathrm{PL_A}} \mathbf A \left[{a_1, \ldots, a_n}\right]$

where $\models_{\mathrm{PL}}$ denotes model of sentence, and $\models_{\mathrm{PL_A}}$ denotes model of formula.

Proof
Applying the Substitution Theorem for Well-Formed Formulas $n$ times:

The result follows from the definitions of model of sentence and $\models_{\mathrm{PL_A}}$ denotes model of formula.