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 the Excluded Middle:
 * $$\vdash p \or \neg p$$

Proof
Let us assume that the Law of the Excluded Middle holds.

Then:
 * $$\vdash p \or \neg p$$
 * $$\vdash q \or \neg q$$

Suppose Peirce's Law did not hold:
 * $$\left({p \implies q}\right) \implies p \not \vdash p$$

Then there exists a model $$\mathcal M$$ such that:

$$ $$

But then:
 * $$\left({F \implies q}\right) \implies F$$

That is:
 * $$F \implies q \vdash F$$

From Implication Properties:
 * $$F \implies q \vdash T$$

From this contradiction we see that it can not be the case that $$\left({p \implies q}\right) \implies p \not \vdash p$$ must be false.

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

that is, Peirce's Law holds.

Now assume that Peirce's Law holds.