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In the Lévy hierarchy, a bounded formula in set theory has only quantifiers of the forms $\forall x\in y$ and $\exists x\in y$. A Sigma-one formula in set theory is an existential unbounded quantifier block followed by a bounded formula in set theory, and a Pi-one formula in set theory has a universal unbounded block. A formula is a Delta-one formula modulo ZFC if ZFC proves it equivalent, with the same free variables, both to a ${\Sigma }_1$ formula and to a ${\Pi }_1$ formula. Equivalence modulo the specified theory is part of the definition; it is not necessary that the original string have both syntactic forms. Such formulas satisfy the usual Delta-one absoluteness between transitive models satisfying the relevant axioms. The modulo-ZF analogue is a Delta-one formula in set theory.

Assume $\phi (\vec{a})$ is a Delta-one formula in set theory. Thus ZF proves it equivalent to a ${\Sigma }_1$ formula $\sigma$ and to a ${\Pi }_1$ formula $\pi$. Only finitely many axioms of ZF occur in these two formal proofs; collect them, together with the finite fragment needed for bounded-formula absoluteness, into $T$.

Let $M$ be a transitive class containing $\vec{a}$ and satisfying $T$. If $M\models \phi$, then $M\models \sigma$, and upward absoluteness of ${\Sigma }_1$ formulas gives $V\models \sigma$, hence $V\models \phi$. If $V\models \phi$, then $V\models \pi$, and downward absoluteness of ${\Pi }_1$ formulas gives $M\models \pi$, hence $M\models \phi$. Therefore ZF proves that $\phi$ is absolute for every such $M$.