Semantic Consequence of Set Union Formula

Theorem
Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

Let $\FF$ be a set of logical formulas from $\LL$.

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

Let $\psi$ be another logical formula.

Then:


 * $\FF \cup \set \psi \models_{\mathscr M} \phi$

that is, $\phi$ is also a semantic consequence of $\FF \cup \set \psi$.

Proof
This is an immediate consequence of Semantic Consequence of Superset.

Also see

 * Semantic Consequence of Superset