Equality is Symmetric

Theorem
Equality is symmetric.

That is:


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

Proof
This proof depends on Leibniz's Law:


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

We are trying to prove $a = b \iff b = a$.

Our assertion, then, is:


 * $(P(a) \iff P(b)) \implies (P(b) \implies P(a))$

From the definition of material equivalence, our assertion is:


 * $(P(a) \implies P(b) \land P(b) \implies P(a)) \implies (P(b) \implies P(a))$

From Rule of Simplification,

$(P(a) \implies P(b)) \land (P(b) \implies P(a)) \vdash P(b) \implies P(a)$

Thus


 * $(P(b) \implies P(a)) \implies (P(b) \implies P(a))$

From Law of Identity, this is a tautology. So the theorem holds.