In first-order set theory, the principle of complete reflection is the following.

$\tag{ CR} \forall \alpha\exists \beta>\alpha\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{x}))$

where $\phi\in \mathcal L_\in$ with free variables among $\vec{x}$ and where $\phi^\beta$ is the result of binding all the quantifiers in $\phi$ to $V_\beta$.

It is well-known that over ZFC minus replacement and infinity, CR implies replacement and infinity. I'm interested in the following principle:

$\tag{ CR*} \forall \alpha\exists \beta>\alpha\exists j\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{j(x)}))$

Essentially, I want to know if CR* implies replacement over the other axioms of ZFC plus the claim that every set is in some $V_\alpha$: that is, over ZFC - replacement + $\forall x\exists \alpha(x\in V_\alpha)$.

However, since $j$ might be a proper class, the question is whether CR* implies first-order replacement in ZFC2 - second-order replacement + $\forall x\exists \alpha(x\in V_\alpha)$.