# Transitive classes and $\Delta_0$- absoluteness.

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 $$M$$.

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.