# Is this form of replacement suitable for ZF - Powerset + well-ordering principle?

The following scheme can be understood as a form of replacement. Axiomatizing $$\sf ZF$$ with it instead of the usual replacement schema renders it immune to removal of extensionality; see here.

In an article by Gitman/Hamkins/Johnstone, it was shown that the usual version of replacement is in some sense problematic if power set axiom is removed, while at the same time having the well-ordering principle.

Would the following version of replacement work as collection does if powerset is removed while having the well-ordering principle?

Replacement$$^*$$: if $$\varphi$$ is a formula in which $$B$$ is not free, then:

$$\forall A \exists B \forall y \, (y \in B \leftrightarrow \exists x \in A: \forall z (z \in y \leftrightarrow \varphi(x,z)))$$

If we have extensionality, then this is:

$$\forall A \exists B \forall y \, (y \in B \leftrightarrow \exists x \in A: y= \{z \mid \varphi(x,z)\})$$

Your version of replacement is a weakening of ordinary replacement. To see this assume that replacement holds (but perhaps not power set or collection), and then observe that for any set $$A$$ and any formula $$\varphi$$, we can first restrict to the subset of $$x\in A$$ for which $$\varphi(x,z)$$ defines a set $$y$$, and then apply replacement to gather these $$y$$s together into a set.