Modus Ponendo Ponens for Semantic Consequence in Predicate Logic

Theorem
Let $\mathrm{PL}$ be the formal semantics of structures for predicate logic.

Denote with $\models_{\mathrm{PL}}$ $\mathrm{PL}$-semantic consequence.

Let $\mathbf A$ and $\mathbf B$ be sentences.

Let $\FF$ be a set of sentences.

Suppose that:


 * $\FF \models_{\mathrm{PL}} \mathbf A$
 * $\FF \models_{\mathrm{PL}} \mathbf A \implies \mathbf B$

Then:


 * $\FF \models_{\mathrm{PL}} \mathbf B$

establishing Modus Ponendo Ponens in $\mathrm{PL}$.

Proof
To establish $\FF \models_{\mathrm{PL}} \mathbf B$, we need to show that for all structures $\AA$:


 * $\AA \models_{\mathrm{PL}} \FF$ implies $\AA \models_{\mathrm{PL}} \mathbf B$

where $\models_{\mathrm{PL}}$ denotes the models relation.

Then since $\AA \models_{\mathrm{PL}} \FF$, it follows that:


 * $\AA \models_{\mathrm{PL}} \mathbf A$
 * $\AA \models_{\mathrm{PL}} \mathbf A \implies \mathbf B$

By definition then, the latter means: