Equivalence of Definitions of Consistent Proof System

Theorem
Let $\mathcal L_0$ be the language of propositional logic.

Let $\mathscr P$ be a proof system for $\mathcal L_0$.

Definition 1 implies Definition 2
Suppose that $\neg\vdash_{\mathscr P} \phi$.

Suppose additionally that there is some logical formula $\psi$ such that:


 * $\vdash_{\mathscr P} \psi, \neg \psi$

By the Rule of Explosion:


 * $\psi, \neg \psi \vdash_{\mathscr P} \phi$

By Provable Consequence of Theorems is Theorem, we conclude:


 * $\vdash_{\mathscr P} \phi$

in contradiction to our assumption.

Definition 2 implies Definition 1
Suppose either $\phi$ or $\neg\phi$ is not a theorem of $\mathscr P$.

The implication follows trivially.