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This is probably a question for Mike, but anybody could answer it.
Have you ever come across an axiom scheme of (say material) set theory, which I’m tentatively calling subset separation (or perhaps better, but longer, power-class-bounded separation), stating the axiom of separation for all formulas in which all quantifiers are guarded by some power class, so of the form $\forall x \subseteq t, \ldots$ or $\exists x \subseteq t, \ldots$, or guarded by a set (as usual).
Subset separation follows from full separation; it also follows from bounded separation and power sets. Arguably, it is the impredicative core common to the axioms of full separation and power sets.
In Mathias’ paper “The strength of Mac Lane set theory”, section 6, he calls formulas of that sort $\Delta^{\mathcal{P}}_0$, so that your axiom scheme would be called “$\Delta^{\mathcal{P}}_0$-separation”. He also uses “restricted” for quantifiers guarded by a set and “limited” for quantifiers guarded by the power class of a set, so that perhaps your axiom could also be called “limited separation” (as opposed to “restricted separation” meaning $\Delta_0$-separation).
He references the following papers, which I have not read:
Thanks!
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