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.