I'm interested in finding the "simplest" formulation of first-order logic. To be precise, what formulation of first-order logic has the fewest total axioms (or axiom schemas, henceforth just called axioms) and inference rules? I'm ok with leaving either $\exists$ or $\forall$ out of the language entirely, since we can define one in terms of the other.
For starters, Appendix C of Robert Wolf's Tour through Mathematical Logic contains seven axioms and one inference rule. In fact, one axiom is the definition of $\exists,$ which I'm ok with omitting, so that leaves just six axioms and one inference rule. Can we do even better?