= 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> $\varphi$, 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
$$
\boxed{U_1\models\varphi(\mathbf a)\iff U_2\models\varphi(\mathbf a)\qquad(\mathbf a\in U_1^n).}
$$
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