Let $\Sigma$ be a signature (decidable, with equality) and $I$ an interpretation structure over said signature. Let $φ$ be a formula. Show $I\Vdash_{\Sigma} φ$ iff $I\Vdash_{\Sigma} \forall x φ$.
This is an exercise in the book Foundations of Logic and Theory of Computation by Amílcar Sernadas. I want this to show that $(\forall x \,\varphi )\models \varphi$ .
[For clarification]
We define $I\Vdash \varphi$ in the following way:
