Show that if $\displaystyle x$ does not occur free in $\displaystyle \alpha$, then $\displaystyle \alpha \models \forall x \alpha$.

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(*)$\displaystyle \models_A \forall x \phi[s]$ iff for every $\displaystyle d \in |A|$, we have $\displaystyle \models_A \phi[s(x|d)]$.

Let $\displaystyle A$ be any structure for the language and $\displaystyle s$ be any function $\displaystyle s: V \rightarrow |A|$ that satisfies $\displaystyle \alpha$. Therefore, $\displaystyle \models_A \alpha[s]$. Since x does not occur free in $\displaystyle \alpha$, $\displaystyle \alpha[s]=\alpha[s(x | d)]$. It follows that $\displaystyle \models_A \alpha[s(x|d)]$. Therefore, $\displaystyle \models_A \forall x \alpha[s]$ by (*). Now the conclusion follows.

Is this proof O.K or am I missing something?

Thank you.