I am confused about the nature of first-order logic's axioms. I want every line of a proof to be a valid sentence, and I want the only rule of inference employed to be modus ponens. What axioms do I need?
How, for example, do I prove $(\forall x\forall y P(x,y))\rightarrow(\forall x P(x,x))$? How about $\forall x (Q(x)\rightarrow Q(x))$?