MHB Proving (x = y) using Axioms: Basic Arithmetic Proof

[agapito]

Which axioms (at minimum) would have to be invoked so the following expression holds:

(x = y) ----> [(y=x) <---> (y=y)] ?

All help appreciated, am

 

[Evgeny.Makarov]

It is not clear what you mean by "hold". If you are referring to the truth of this formula in the standard interpretation for natural numbers, then no axioms are involved: any closed formula is simply either true or false. If you are referring to derivability of this formula, then you need to specify the theory from which you are deriving, e.g., theory of equality. I believe the left-to-right direction can be proved using Leibniz's law, and the right-to-left direction also requires reflexivity.