I am just finishing teaching a course in Set Theory, and was thinking about the ZFC axioms, when the following axiom occurred to me:
Axiom: Given a sentence $S$ with free variables among $x,y,A_1,\ldots,A_k$
$\forall x\in X, \exists y$ such that $S(x,y,A_1,\ldots,A_k)$
then there exists a function $f$ with domain $X$ such that
$\forall x\in X, S(x,f(x),A_1,\ldots,A_k)$.
(I guess this is really an axiom schema.)
This seems to imply the axiom of choice: Take $k=1$, let $A_1$ be a family of nonempty sets and let $X$ be the indexing set of the family. Let $S$ be the sentence that says that $y$ is in the set indexed by $x$. Then the function guaranteed by the Axiom is a choice function for the family. This also seems to imply the Axiom of Replacement, too, and in fact the new Axiom seems to be equivalent to Replacement plus Choice.
I like the Axiom because it says something I find very intuitive: If you know that for all x there exists a y such that BLAH, you can find a function sending each x to a y such that BLAH.
This is probably not of great importance, but I thought I would ask, in case anyone is aware of an approach that combines the axioms like this.