Are you doing the Henkin proof? If so, isn't it true that the sentence

is a first order validity? I suppose you can prove this if you really need to - the natural deduction rules are strong enough to get it.

So my question is, why do you need to prove this? In the Henkin construction, at least, you usually just take these statements as axioms. At least, the bookLanguage, Proof, and Logicdoes it that way. What book are you using?