This is a follow-up to this question.
Let $\text{ZFC}^-$ be ZFC without powerset and with collection rather than replacement, as described here.
Is there a $\Pi_2$ (or perhaps $\Sigma_2$) sentence $A$ such that $\text{ZFC}^- + A$ proves powerset and is consistent?
No, using downward Löwenheim-Skolem theorem (and transitive collapses), every true $Π_2$ statement holds in $H(λ)$ for every cardinal $λ>ω$. ($H(λ)$ consists of all sets whose transitive closure has cardinality $<λ$.) Moreover, every true $Σ_3$ statement holds in every $H(λ)$ large enough to contain a witness for the statement. Thus, power set is not provable in ZFC\P from a consistent $Σ_3$ statement.
