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 $\mathrm{PL}$.