Question about deduction in first order logic

Question

My question is: Assume that $\Gamma\vdash\varphi$ and that $P$ is a predicate symbol which occurs neither in $\Gamma$ nor in $\varphi$. Is there a deduction of $\varphi$ from $\Gamma$ in which $P$ doesn't occur?

My textbook suggests two possible approaches: one approach which makes use of two languages (one that contains $P$ and one that does not), and the other approach which considers the question whether $P$ can be systematically eliminated from a given deduction $\varphi$ from $\Gamma$.

I am really confused about where to start with this problem...how do I construct the two languages to use?

Answer 1

Replace $P$ in every step of the proof with some other predicate variable $Q$ in the language (or even by a predicate constant like $\top$ or $\bot$), and the result will still be a valid proof which does not use $P$.

Answer 2

Whatever deductive system you are using, you have presumably proven Soundness and Completeness theorems for it. Then having a deduction in language $\mathcal{L}$ is equivalent to being logically entailed among $\mathcal{L}$-structures. What happens on the logical entailment side when you throw away the predicate symbol?