 Solution

Begin with bounded formulas in set theory. Atomic membership and equality have the same interpretation on shared elements of $U_1\subseteq U_2$, each a transitive class. Boolean connectives preserve agreement. For a bounded quantifier with bound $a\in U_1$, every member of $a$ lies in $U_1$ by transitivity. Its quantifier range is therefore exactly the same in both structures. Structural induction proves set-theoretic absoluteness for every bounded formula in set theory.

A Sigma-one formula in set theory is upward absolute: witnesses in $U_1$ are still witnesses in $U_2$, and the bounded matrix agrees. A Pi-one formula in set theory is downward absolute, by negation or by restricting its universal quantifiers.

For the given Delta-one formula modulo ZF $\phi$, select a ${\Sigma }_1$ equivalent $\sigma$ and a ${\Pi }_1$ equivalent $\pi$. Both structures satisfy ZF, so both equivalences hold in both structures. Upward absoluteness of $\sigma$ proves the forward implication, and downward absoluteness of $\pi$ proves the reverse implication. Thus Delta-one absoluteness gives