Equality is Reflexive

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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.

$\blacksquare$


Sources