This is probably a dumb question, but I'm stuck at a step of the proof of the Condensation Lemma in Devlin's Constructibility.
Context: we have $X\prec_1 L_\alpha$ for some limit $\alpha$ and want to show that the Mostowski collapse of $X$ is $L_\beta$ for some $\beta\le\alpha$.
Having proven before that the formula $v=L_\gamma$ is $\Delta_1^{KP}$, Devlin says that there is a $\Sigma_0$ formula $\Phi(z,v,\gamma)$ of LST such that
$$ \forall\gamma\forall v[v=L_\gamma \leftrightarrow\exists z \Phi(z,v,\gamma)]$$ So far so good. He then claims that if $\varphi(z,v,\gamma)$ is the $\mathscr{L}$-analogue of $\Phi(z,v,\gamma)$ (i.e. $\mathscr{L}$ is the formal version of LST, so $\Phi$ is a metatheoretic formula, and $\varphi$ is a set representing that formula), then
$$ \forall\gamma<\alpha\forall v[v=L_\gamma \leftrightarrow v\in L_\alpha \wedge L_\alpha\models \exists z \varphi(z,v,y)]$$
I'm confused by the $\to$ direction, namely how is he going from $\exists z\Phi(z,v,\gamma)$ to $L_\alpha\models\exists z\varphi(z,v,\gamma)$? That is to say, why is there a $z$ with $\Phi(z,v,\gamma)$ in $L_\alpha$. The ordinal $\alpha$ is merely a limit, not necessarily admissible, so appealing to $\Delta_1^{KP}$ absoluteness seems shady. I assume I'm missing something obvious, but I can't see it right now.
Thanks!