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.