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Fix a logical formula $\theta (x,y,\vec{p})$ with parameters in $V$, and a set $a\in V$. Suppose $⟨V,\in ⟩$ satisfies that each $x\in a$ has a unique $y$ with $\theta (x,y,\vec{p})$. Relativize every unbounded quantifier in $\theta$ to the definable class in set theory $V$, obtaining an ambient first-order formula ${\theta }^V$.

Transitivity of $V$ ensures that every member of $a$ belongs to $V$. The ambient relationtherefore defines a unique ambient set $y$ for every $x\in a$. Apply ambient replacement to obtain its range set $b$. By construction every element of $b$ belongs to $V$, so part (ii) gives $b\in V$.

Membership in this set $b$ is exactly the range relation required by replacement inside the structure $⟨V,\in ⟩$. Since the logical formula and parameters were arbitrary, every instance of the replacement scheme holds in the hierarchy class in set theory. The proof uses formula relativization to a class and the closure property from part (ii), not the unsupported assertion that $V$ is the whole ambient universe.

For parameters $a_1,\dots ,a_n\in M$, set-theoretic absoluteness means agreement of the two membership first-order structures:This must hold for every such parameter tuple. Equivalently, formula relativization to a class produces two formulas ${\phi }^M$ and ${\phi }^N$ whose truth values agree on tuples from $M$. Agreement for parameters only in $N\setminus M$ is not part of the definition. For a proper class in set theory, this notation is understood separately for each fixed first-order formula, rather than as an unrestricted truth predicate for the entire universe.

Conversely, suppose a finite $T\subseteq \mathrm{ZF}$ has the stated absoluteness property. LetBecause $T$ 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

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 $\phi$. Consequently ZF provesThus $\phi$ is both ${\Sigma }_1^{\mathrm{ZF}}$ and ${\Pi }_1^{\mathrm{ZF}}$, so it is ${\Delta }_1^{\mathrm{ZF}}$.

Takethe formula relativization to a class: replace each $\exists y\psi$ byand each $\forall y\psi$ byleaving atomic formulas unchanged. If $M$ is given by a defining formula, insert that formula wherever $y\in M$ occurs; this is why the map may depend on $M$.

For $z$ and any parameters in $M$, induction on formulas givesTherefore ${\phi }^{\ast }$-closure putsin $M$ for every $x\in M$. This is exactly the relativized closure criterion for separation, so $(M,\in )$ satisfies the $\phi$-instance of separation.