Semantic Consequence of Superset

Theorem
Let $\mathcal L$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\mathcal L$.

Let $\mathcal F$ be a set of logical formulas from $\mathcal L$.

Let $\phi$ be an $\mathscr M$-semantic consequence of $\mathcal F$.

Let $\mathcal F'$ be another set of logical formulas.

Then:


 * $\mathcal F \cup \mathcal F' \models_{\mathscr M} \phi$

that is, $\phi$ is also a semantic consequence of $\mathcal F \cup \mathcal F'$.

Proof
Any model of $\mathcal F \cup \mathcal F'$ is a fortiori also a model of $\mathcal F$.

By definition of semantic consequence all models of $\mathcal F$ are models of $\phi$.

Therefore all models of $\mathcal F \cup \mathcal F'$ are also models of $\phi$.

Hence:


 * $\mathcal F \cup \mathcal F' \models_{\mathscr M} \phi$

as desired.