Peirce's Law is Equivalent to Law of Excluded Middle

Theorem
Peirce's Law:
 * $\left({p \implies q}\right) \implies p \vdash p$

is logically equivalent to the Law of Excluded Middle:
 * $\vdash p \lor \neg p$

That is, Peirce's Law holds iff the Law of Excluded Middle holds.

Law of Excluded Middle implies Peirce's Law
Let the truth of the Law of Excluded Middle be assumed.

Then:
 * $\left({p \lor \neg p}\right) \vdash \left({\left({p \implies q}\right) \implies p}\right) \implies p$

is demonstrated, as follows.

The result follows by an application of Modus Ponendo Ponens:


 * $\left({\left({p \implies q}\right) \implies p}\right) \implies p, \left({p \implies q}\right) \implies p \vdash p$