Proof that $x \neq x$ is false, in logic

Question

We know from the reflexivity axiom that $x = x$ is true by definition.

So how do I show that $x \neq x$ is false? I know "intuitively" it's false but I don't know how to "show" it. I'm not sure what kind of truth table I need to build or if I need to state equality in some other way so I can work with it.

Do I need to reframe equality as the "if and only if" truth table?

Answer 1

It depends how you define $\neq$, but I will adopt the definition that $x\neq y$ iff $\neg(x= y)$, where $\neg$ negates a proposition. Since $p\to \neg(\neg p)$, the consequent being the falsity of $\neg p$, we're done. Note this argument only requires intuitionistic logic.