Semantic Consequence of Set minus Tautology

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 $\psi \in \FF$ be a tautology.

Then:


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

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

Proof
Let $\MM$ be a model of $\FF \setminus \set \psi$.

Since $\psi$ is a tautology, it follows that:


 * $\MM \models_{\mathscr M} \psi$

Hence:


 * $\MM \models \FF$

which,, entails:


 * $\MM \models \phi$

Since $\MM$ was arbitrary, it follows by definition of semantic consequence that:


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