Equality is Reflexive

Theorem
Equality is reflexive.

That is:
 * $\forall a: a = a$

Proof
This proof depends on Leibniz's law:


 * $x = y \dashv \vdash \map P x \iff \map P y$

We are trying to prove $a = a$.

Our assertion, then, is:


 * $a = a \dashv \vdash \map P a \iff \map P a$

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

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

Also see

 * Equality is Symmetric
 * Equality is Transitive
 * Equality is Equivalence Relation