= Solution
Conversely, suppose a finite $T\subseteq\mathrm{ZF}$ has the stated absoluteness property. Let
$$
\sigma(\vec a)\equiv\exists N\,[N\text{ is a transitive set},\ \vec a\in N,\ N\models T,\ N\models\varphi(\vec a)].
$$
Because $T$ is finite, every satisfaction assertion here can be replaced by the corresponding <formula relativization to a class>. All quantifiers in the matrix are bounded by $N$, so $\sigma$ is $\Sigma_1$. Define the $\Pi_1$ formula
$$
\pi(\vec a)\equiv\neg\exists N\,[N\text{ is a transitive set},\ \vec a\in N,\ N\models T,\ N\models\neg\varphi(\vec a)].
$$
By the <Lévy reflection theorem>, ZF proves that for any parameters $\vec a$ there is a level $V_\alpha$ containing them and satisfying the finite fragment $T$. The assumed absoluteness says that every such transitive set agrees with $V$ about $\varphi$. Consequently ZF proves
$$
\varphi(\vec a)\leftrightarrow\sigma(\vec a)
\qquad\text{and}\qquad
\varphi(\vec a)\leftrightarrow\pi(\vec a).
$$
Thus $\varphi$ is both $\Sigma_1^{\mathrm{ZF}}$ and $\Pi_1^{\mathrm{ZF}}$, so it is $\Delta_1^{\mathrm{ZF}}$.
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