Say we're given that $S$ and $S'$ are representable relations definable in $Th(A_E)$ by $\psi$ and $\psi'$ respectively and $\psi$ and $\psi'$ have the property that $Th(A_E) \vdash \forall y (\exists x \psi(x,y) \leftrightarrow \forall x' \psi'(x', y))$. I would like to show that the unary relation defined by $\exists x \psi(x,y)$ is representable in $T_E$.
Here's what I've tried: We call our new relation $D$ which we will proceed to show is representable in $Th(A_E)$. To do so, we must show that for every $a \in \mathbb{N}$, $\langle a \rangle \in D$ iff $\vDash_\mathcal{N} \exists x \psi(x,a)$. We see that the latter statement holds when we have a proof of it. Here's where I get stuck and don't see how to use $S$ and $S'$... I don't know what sort of proof would utilize $\psi$ and $\psi'$. Any tips would be much appreciated.