The following lemma was used to prove the Łos-Tarski theorem:
Lemma. Let $T$ be a theory and let $\phi(x)$ be a formula in which the constant $c$ does not occur. Then $T\models\phi(c)$ if and only if $T\models\forall x\phi(x)$.
From right to left the statement is trivial. How can one prove the other direction? In particular, how can one use the assumption that $c$ does not occur in $\phi(x)$?
In short, if $c$ does not occur in $T$ we can choose a "fresh" variable $y$ and replace everywhere in the proof $T⊢\phi(c)$ the constant $c$ with the variable $y$.
The result will be a valid proof $T⊢\phi(y)$.
Then we apply the Generalization Theorem to conclude wirh $∀y \ \phi(y)$: we can do it because $y$ does not occur in $T$.
From a "semantical" point of view, if $c$ is not in the set of "axioms" $T$, we have that the "axioms" do not put any constraint on the meaning of $c$.
This means that, if $T⊨ \phi(c)$, then every interpretation that satisfy the "axioms" will also satisfy $\phi(c)$, whatever is the object of the domain of the interpretation that we choose to assign to the constatnt $c$ as its denotation.