Theory of Set of Formulas is Theory

Theorem

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

Let $\FF$ be a set of $\LL$-formulas.

Let $\map T \FF$ be the $\LL$-theory of $\FF$.

Then $\map T \FF$ is a theory.

Let $\phi$ be a $\LL$-formula.

Suppose that $\map T \FF \models_{\mathscr M} \phi$.

By definition of $\map T \FF$:

$\FF \models_{\mathscr M} \psi$

for every $\psi \in \map T \FF$.

Hence by definition of semantic consequence:

$\FF \models_{\mathscr M} \map T \FF$

$\FF \models_{\mathscr M} \phi$

Finally, by definition of $\map T \FF$:

$\phi \in \map T \FF$.

Since $\phi$ was arbitrary, the result follows.

$\blacksquare$