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 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:


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