I want to see if the following result is correct or not:
Let $\varphi(x)$ be a formula in $\mathcal{L}_{\mathrm{ZF}}$ with only bounded quantifiers such that $\exists x\,\varphi(x)$ holds in $V$, then $\exists x\,\varphi(x)$ holds in $H_{\omega_1}$.
I wrote a proof for it using the following reasoning:
$\varphi$ is absolute for all transitive models; let $\exists x\,\varphi(x)$ be true, then $\varphi(a)$ holds for some $a$. Let $\vartheta\in\mathbf{ON}$ such that $a\in V_\vartheta$. Let $M\prec V_\vartheta$ be a countable elementary substructure, $\bar M$ be the Mostowski collapse ($\bar M\cong M$), then $\bar M$ is transitive and countable, hence hereditarily countable, so $\bar M\subseteq H_{\omega_1}$, so there is a witness for $\varphi(x)$ in $H_{\omega_1}$.
My questions are: (1) is the result above correct? (2) Is the proof given correct? (3) If the result is correct and the proof is not correct, what would be a correct proof for the result?