Exercise 17, chapter IV in Kunen's 1980 "Set Theory" reads:
"Show that for any formula $\phi$, the following are equivalent:
- $\phi$ is $\Delta_1^{ZF}$.
- There is a finite set $S$ of axioms in $ZF$ such that: $$ZF \vdash \forall M\left(M \text{ transitive } \wedge \left(\bigwedge_{\sigma \in S}\sigma^M \right) \to \phi \text{ is absolute for } M \right) $$
Hint: For $(2)\implies (1)$, use reflection."
I can prove $(1)\implies (2)$ by proving $\phi$ is $\Delta_1^M$ and hence both downward and upward absolute. But I cannot figure out the other direction. I can use Reflection Theorem to prove there are transitive $M$ satisfying $\bigwedge_{\sigma \in M}\sigma^M$ but that doesn't seem to help.
One of the bottleneck is I have no idea how to prove a formula $\phi$ is $\Delta_1^{ZF}$ without knowing exactly what $\phi$ is. Any suggestion is appreciated.