In ordinary membership-based set theory, the axiom schema of replacement states that if $\phi$ is a first-order formula, and $A$ is a set such that for any $x\in A$ there exists a unique $y$ such that $\phi(x,y)$, then there exists a set $B$ such that $y\in B$ if and only if $\phi(x,y)$ for some $x\in A$. That is, $B$ is the "image" of $A$ under the "definable class function" $\phi$.
The related axiom schema of collection modifies this by not requiring $y$ to be unique, but only requiring $B$ to contain some $y$ for each $x$ rather than all of them. However, there are at least two different versions of this.
If for all $x\in A$ there exists a $y$ with $\phi(x,y)$, then there exists a set $B$ such that for all $x\in A$ there is a $y\in B$ with $\phi(x,y)$ (this is Wikipedia's version; I'll call it "weak collection").
If for all $x\in A$ there exists a $y$ with $\phi(x,y)$, then there exists a set $B$ such that (1) for all $x\in A$ there is a $y\in B$ with $\phi(x,y)$, and (2) for all $y\in B$ there is an $x\in A$ with $\phi(x,y)$ (I'll call this "strong collection").
The third possibly relevant axiom is the axiom schema of separation, which states that for any $\phi$ and any set $A$ there exists a set $B\subseteq A$ such that $x\in B$ if and only if $x\in A$ and $\phi(x)$.
I know the following implications between these axioms:
- Strong collection implies weak collection, since it has the same hypotheses and a stronger conclusion.
- Strong collection implies replacement, since it has a weaker hypothesis and the same conclusion.
- Replacement implies separation (assuming excluded middle): apply replacement to the formula "($\phi(x)$ and $y=\lbrace x\rbrace$) or ($\neg\phi(x)$ and $y=\emptyset$)" and take the union of the resulting set.
- Together with AC and foundation, replacement implies weak collection: let $\psi(x,V)$ assert that $V=V_\alpha$ is the smallest level of the von Neumann hierarchy such that there exists a $y\in V_\alpha$ with $\phi(x,y)$, apply replacement to $\psi$ and take the union of all the resulting $V_\alpha$.
- Weak collection and separation together imply strong collection: separation cuts out the subset of $B$ consisting of those $y$ such that $\phi(x,y)$ for some $x\in A$.
My question is: does weak collection imply replacement (and hence also separation and strong collection) without assuming separation to hold a priori? Feel free to assume all the other axioms of ZFC . (including $\Delta_0$-separation). I'm fairly sure the answer is "no," but several sources I've read seem to assume that it does. Can someone give a definitive answer, and ideally a reference?