# Equality is Transitive

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## Theorem

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

## Proof

 $\displaystyle a$ $=$ $\displaystyle b$ $\displaystyle \vdash \ \$ $\displaystyle \map P a$ $\iff$ $\displaystyle \map P b$ Leibniz's law $\displaystyle b$ $=$ $\displaystyle c$ $\displaystyle \vdash \ \$ $\displaystyle \map P b$ $\iff$ $\displaystyle \map P c$ Leibniz's law $\displaystyle \vdash \ \$ $\displaystyle \map P a$ $\iff$ $\displaystyle \map P c$ Biconditional is Transitive $\displaystyle \vdash \ \$ $\displaystyle a$ $=$ $\displaystyle c$ Leibniz's law

$\blacksquare$