Lemma on constants

The Lemma on Constants is an elementary result, which says something like the following:

Lemma: Suppose $$T \vdash \phi(c)$$, where $$T$$ is a theory, $$\phi(x)$$ is a formula, x is a tuple of variables, and c is a tuple of constant symbols not appearing in T. Then $$T \vdash \forall x : \phi(x)$$.

Proof: If one interprets $$T \vdash \psi$$ to mean that $$\psi$$ holds in all models of T, then this follows from unwinding the definitions: $$T \vdash \phi(c)$$ means that whenever M is a model of T and c is a tuple from M, $$M \models \phi(c)$$. Then clearly $$M \models \forall x: \phi(x)$$!

On the other hand, if one interprets $$T \vdash \psi$$ to mean that $$\psi$$ can be proven from T, then this follows from the other version by Gödel's Completeness Theorem, which says that these two interpretations of $$\vdash$$ are the same. QED.

The Lemma on Constants is practically obvious. However, it is handy to have a name for this lemma when making technical arguments.