Choice Collection: if $\phi(x,y)$ is a formula in which "$B$" doesn't occur free, and "$x,y$" are among its free variables; then: $$\forall A \exists B \forall x \in A \, (\exists y \, \phi(x,y) \implies \exists! y : \langle x, y \rangle \in B \land \phi(x,y))$$
$\langle ,\rangle$ stands for ordered pair.
We shall denote this axiom by "$\sf C.Col$".
So $\sf ZFC$ can be written as $\sf Z + C.Col$
This axiom would prove both $\sf AC$ and Collection\Boundedness schema over the rest of axioms of $\sf Z$. It is a theorem of $\sf ZFC$ and $\sf NBG$ \ $\sf MK$.
Can $\sf Z+ C.Col-Power$ prove that every set can be well ordered?