Semantic Consequence is Transitive

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

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

Let $\mathcal F, \mathcal G$ and $\mathcal H$ be sets of $\mathcal L$-formulas.

Suppose that:


 * $\mathcal F \models_{\mathscr M} \mathcal G$
 * $\mathcal G \models_{\mathscr M} \mathcal H$

Then $\mathcal F \models_{\mathscr M} \mathcal H$.

Proof
Let $\mathcal M$ be an $\mathscr M$-structure.

By assumption, if $\mathcal M$ is a model of $\mathcal F$, it is one of $\mathcal G$ as well.

But any model of $\mathcal G$ is also a model of $\mathcal H$.

In conclusion, any model of $\mathcal F$ is also a model of $\mathcal H$.

Hence the result, by definition of semantic consequence.