Semantic Consequence of Set minus Tautology

Theorem

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

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

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 $\FF \setminus \set \psi$.

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, by hypothesis, entails:

$\MM \models \phi$

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

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

$\blacksquare$