# Axiom:Axioms of Equality

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## 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

- $\forall a, b, c: \paren {a = b} \land \paren {b = c} \implies a = c$