Equality is Reflexive

Theorem
Equality is reflexive. That is,

$\forall a: a = a$.

Proof
This proof depends on Leibniz's Law:

$(1) \quad x = y \dashv \vdash P(x) \implies P(y) \land P(y) \implies P(x)$

We are trying to prove $a = a$. By $(1)$, our assertion is

$(2) \quad P(a) \implies P(a) \land P(a) \implies P(a)$

It is sufficient to show that this statement is interderivable with a tautology.

Statement $(2)$ has the form the form $p \land p$. From the Rule of Idempotence we can write

$(3) \quad P(a) \implies P(a)$

Statement $(3)$ has the form $p \implies p$. From Law of Identity this is a tautology. Thus,

$x = x \dashv \vdash \top$