Here's my attempt so far in proving the equivalence:

The question says "x does not occur freely in A" (which means it is either the variable immediately following a quantifier or it is in the scope of some quantifier involving x). Let I be an interpetation with domain on D, and v be an I-assignment. Suppose $\displaystyle I \models_v \exists x (A \vee B)$. We wish to show that $\displaystyle I \models_v (A \vee \exists x B)$. I don't know if this is correct but I'm considering two cases depending on wether $\displaystyle A^I(d)$ holds for all $\displaystyle d \in D$:

Case 1: $\displaystyle A^I(d)$ holds for all $\displaystyle d \in D$. So $\displaystyle I \models_{v \frac{d}{x}} A$ so $\displaystyle I \models_{v \frac{d}{x}} (A \vee \exists x B)$. Is this correct?

Case 2: There is some $\displaystyle d \in D$ such that $\displaystyle A^I(d)$ does not hold... And we assumed that there is a d such that $\displaystyle I \models_{v\frac{d}{x}} (A \vee B)$. But how can I complete this part?

I greatly appreciate any help on how to prove this question...