# Equality is Symmetric

## Theorem

That is:

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

## Proof

 $\displaystyle a$ $=$ $\displaystyle b$ $\displaystyle \vdash \ \$ $\displaystyle P \left({a}\right)$ $\iff$ $\displaystyle P \left({b}\right)$ Leibniz's Law $\displaystyle \vdash \ \$ $\displaystyle P \left({b}\right)$ $\iff$ $\displaystyle P \left({a}\right)$ Biconditional is Commutative $\displaystyle \vdash \ \$ $\displaystyle b$ $=$ $\displaystyle a$ Leibniz's Law

$\blacksquare$