Equality is Symmetric

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Theorem

Equality is symmetric.


That is:

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


Proof

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

$\blacksquare$


Sources