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On page 99, A Mathematical Introduction to Logic, Herbert B. Enderton(2ed),

Show that if $x$ does not occur free in $α$, then $α \vDash ∀ x α$.

Added:This could be seen as a follow-up problem of this one. Frankly speaking, I'm totally confused between this problem and the answer to that problem.

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up vote 1 down vote accepted

"$\vDash$" means that RHS is true in every model of LHS. If $x$ does not occur free in $\alpha$, a model of $\alpha$ need not assign a value to $x$ - that is, in a model of $\alpha$, $\alpha$ is true for all assignments of $x$, so $\alpha \vDash \forall x \alpha$.

I'm not really sure if this is how formally you are expected to answer something like this (assuming this is attached to some kind of course in logic), but this is the reason it's true.

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