Axiom:Axioms of Equality

Theorem

The axioms of equality are strictly speaking not axiomatic at all, as they can be deduced from still more basic axioms, in particular Leibniz's Law:

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

where $P \left({x}\right)$ and $P \left({y}\right)$ are propositional functions on the elements $x$ and $y$ of the universe of discourse.

Equality is Reflexive

$\forall a: a = a$

Equality is Symmetric

$\forall a, b: a = b \implies b = a$

Equality is Transitive

$\forall a: a = a$