Transitive classes and $\Delta_0$- absoluteness.

Question

I want to show that in $L_{\epsilon}$, formulas in $\Delta_0$ are absolute, that is $$\phi \leftrightarrow \phi^M$$ where $M$ is a transitive class and $\phi^M$ denotes the relativization of $\phi$ to $M$. The part that I am struggling is the direction $\phi \rightarrow \phi^M$ when $\phi = \exists \, x \in y \, \psi(x,y, \dots)$. The relativization of this is $$\phi^M = \exists \, x \in M \left( x \in y \land \psi^M\right) \, .$$ Using the induction hypothesis, $\psi^M \leftrightarrow \psi$ and therefore I can write $$\exists \, x \left( x \in y \land \psi\right)$$ but I cannot add the $x \in M$ since I don't know whether $y \in M$, otherwise from transitivity I could conclude that $x$ is indeed in $M$.

Answer 1

First note that $y$ is a free variable. So for $\phi\leftrightarrow\phi^M$ to even make sense, we have to assign $y$ the value of some $m\in M$.

But now $M$ is transitive, so if $x\in y$, then $x\in M$ as well.