first order: show that $\forall x (\phi \vee \psi) =\hspace{-.4em}|\models (\phi \vee \forall x \psi)$

Question

first order: show that $\forall x (\phi \vee \psi) =\hspace{-.4em}|\models (\phi \vee \forall x \psi)$ if $x \notin free(\phi)$, where $=\hspace{-.4em}|\models$ denotes logically equivalent.

I don't see why these are logically equivalent. It must hinge on x being bounded

Answer 1

Think of ϕ as a constant as far as the variable x, and the quantifier are concerned. The quantifier does no work in ϕ, so excluding ϕ from the quantifier's scope doesn't affect the truth of the the entire formula. So both formulae are logically equivalent.