Equality is Reflexive

Theorem
Equality is reflexive.

That is:


 * $\forall a: a = a$

Proof
This proof depends on Leibniz's Law:


 * $x = y \dashv \vdash P \left({x}\right) \iff P \left({y}\right)$

We are trying to prove $a = a$.

Our assertion, then, is:


 * $a = a \dashv \vdash P(a) \iff P(a)$

From Law of Identity, $P(a) \iff P(a)$ is a tautology.

Thus $a=a$ is also tautologous, and the theorem holds.