# Semantic Consequence of Set Union Formula

## 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 $\psi$ be another logical formula.

Then:

$\mathcal F \cup \left\{{\psi}\right\} \models_{\mathscr M} \phi$

that is, $\phi$ is also a semantic consequence of $\mathcal F \cup \left\{{\psi}\right\}$.

## Proof

This is an immediate consequence of Semantic Consequence of Superset.

$\blacksquare$