[I'll use 'A' and 'E' for the universal and existential quantifiers, respectively, and 'e' for 'is a member of'.]

(1) EuAz(Ex(xeC & zex) -> zeu)

and

(2) EuAz(Ex(xeC & zex) <-> zeu)

With an instance of the axiom schema of separation, we can derive (2) from (1):

EuAz(Ex(xeC & zex) -> zeu).

So let Az(Ex(xeC & zex) -> zeu).

Separation gives us EvAz((zeu & Ex(xeC & zex)) <-> zev).

Let Az((zeu & Ex(xeC & zex)) <-> zev).

So Az(Ex(xeC & zex) <-> zev).

So, generalizing 'v' to 'u', we get EuAz(Ex(xeC & zex) <-> zeu).

The difference then is just a slight technical detail.