Propositional Tautology is Tautology in Predicate Logic

Theorem
Let $\LL_1$ be the language of predicate logic.

Let $\LL_0$ be the language of propositional logic.

Let the basic WFFs of $\LL_1$ be the vocabulary of $\LL_0$.

Let $\mathbf A$ be a $\mathrm{BI}$-tautology of $\LL_0$, by considering the basic subformulas of $\mathbf A$ as part of the vocabulary of $\LL_0$.

Then $\mathbf A$ is a $\mathrm{PL_A}$-tautology of $\LL_1$.

Proof
We proceed by the Principle of Structural Induction on the bottom-up specification of $\LL_1$, applied to $\mathbf A$.

Define $\map {v_F} {\mathbf A} = F$ for all basic WFFs $\mathbf A$ as a boolean interpretation for $\LL_0$.

Consider the case $\mathbf A$ is formed by either $\paren{ \mathbf W ~ \PP_n }$ or $\paren{ \mathbf W ~ \forall, \exists }$.

Then $\mathbf A$ is basic, and therefore:


 * $\map {v_F} {\mathbf A} = F$

meaning that $\mathbf A$ is not a $\mathrm{BI}$-tautology.

Consider next the case $\mathbf A$ is formed by either $\paren{ \mathbf W ~ \neg }$ or $\paren{ \mathbf W ~ \lor, \land, \Rightarrow, \Leftrightarrow }$.

That is, $\mathbf A = \neg \mathbf B$ or $\mathbf A = \mathbf B \circ \mathbf B'$ for $\circ$ being one of $\lor, \land, \Rightarrow, \Leftrightarrow$.

Suppose $\mathbf A$ is not a $\mathrm{PL_A}$-tautology.

Then there are a structure $\AA$ and an assignment $\sigma$ such that:


 * $\AA, \sigma \not\models \mathbf A$

Then define the boolean interpretation $v_{\AA, \sigma}$ on basic WFFs $\mathbf B$ by:


 * $\map {v_{\AA, \sigma} } {\mathbf B} = T$ $\AA, \sigma \models \mathbf B$

which is to say:


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

where $\map {\operatorname{val}_\AA} {\mathbf B} \sqbrk \sigma$ is the value of $\mathbf A$ in $\AA$ under $\sigma$.

Hence, by definition of boolean interpretation:

And similarly, for $\circ$ being one of $\lor, \land, \Rightarrow, \Leftrightarrow$:

Since by hypothesis $\AA, \sigma \not\models \mathbf A$, it follows that:


 * $\map {v_{\AA, \sigma} } {\mathbf A} = F$

and $\mathbf A$ is not a $\mathrm{BI}$-tautology.

The induction case for $\paren{ \mathbf W ~ \neg }$ and $\paren{ \mathbf W ~ \lor, \land, \Rightarrow, \Leftrightarrow }$ follows by transposition.

Hence the result by the Principle of Structural Induction.