I have been trying to prove (for my own entertainment) that XOR is associative.
However, having reduced $(p \oplus q ) \oplus r = p \oplus (q \oplus r)$ to canonical form, so that the only logical operations are OR, AND; I end up having something that I cannot reduce without relying on the associative and distributive properties of AND and OR.
It seems rather redundant to prove that one logical operator has a certain property but rely on another to do so.
My question is: Is the associative property of XOR provable algebraically - or is it axiomatic? Does the truth table count as a proof?
EDIT: This is the table I came up with.