Equivalent forms of other axioms in ZFC? I've heard this joke thrown around: 

The Axiom of Choice is obviously true, the well-ordering principle is obviously false, and who can guess about Zorn's lemma?

Of course, all three statements are equivalent. Are there non trivial reformulations of any of other axioms for ZFC?
 A: An interesting reformulation of ZF uses levels (the levels of the Von Neumann hierarchy, actually). We have of variables two types: sets ($x$,$y$,...) and levels ($V$, $V'$,...). Can be proved that ZF is equivalent to Extensionality, Separation, Infinity, Replacement and:
Accumulation:
$$\forall V'\forall x[x\in V'\iff\exists V\in V'(x\in V\vee x\subset V)].$$
Restriction:
$$\forall x\exists V(x\subset V).$$
(exercise 3.11.9 of Set Theory by Frank Drake)
A: Sure, as many as you like. For example:


*

*Relative to the other axioms of $\operatorname{ZFC}$, the axiom scheme of replacement and the axiom scheme of collection are equivalent. However, this is no longer true, when dropping the power set axiom.

*$\operatorname{ZFC}$ is equivalent to $\operatorname{ZFC}^0$, where the latter doesn't allow parameters in its axiom schemes. See here.

*There are, relative to the other axioms of $\operatorname{ZFC}$, many equivalent formulations of the axiom of infinity, e.g. "$V_{\omega}$ exists".

*The usual formulation of $\operatorname{ZFC}$ is redundant. Therefore you can leave out any of the redundant axioms, say $\phi$. Then, relative to the other axioms of $\operatorname{ZFC}$, $\phi$ is equivalent to $\psi$ for any $\psi$ with $\operatorname{ZFC} \vdash \psi$.

*...

A: Something  that I have read of, but have not studied: Take the versions of the Comprehension and Separation schemas in Kunen [1]. Kurt Godel showed that in ZF minus Comprehension, the Comprehension schema can be replaced by 8 sentences that he presented (or,equivalenty, by one cumbersome sentence.)  I dk if Infinity matters here but I doubt it.
Reference: [1].Kunen, K. Set Theory: An Introduction to Independence Proofs. 
