Clavius's Law implies Law of Excluded Middle

Theorem
Taking Clavius's Law:
 * $\neg p \implies p \vdash p$

as a rule of logic we can prove Law of Excluded Middle:
 * $\vdash p \lor \neg p$

Remark
The specific form of De Morgan's Laws used in this proof does not itself rely on Law of Excluded Middle in any way.