Maybe it will be helpful if you think about it from a semantic point of view. Suppse that if $\Gamma\models\varphi$, and a model ($\mathcal{M},v)\models\Gamma$. Then we have $(\mathcal{M},v)\models\varphi$. Our goal is to show $(\mathcal{M},v)\models\forall_x\varphi$. We also know that $\varphi$ is a axiom, or an elemnent of $\Gamma$, or got from $MP$ from a syntactic point of view.
Suppose is $\varphi$ is an axiom, of course, $(\mathcal{M},v)\models\forall_x\varphi$.
Suppose is $\varphi$ an elemnent of $\Gamma$, or got from MP, then we also have $\mathcal({M},v)\models\forall_x\varphi$. Because we can prove that: if x is not a free variable of $\varphi$, $(\mathcal{M},v)\models\varphi\leftrightarrow(\mathcal{M},v)\models\forall_x\varphi$ by induction on the compexity of $\varphi$.