Proof by Contradiction for Hilbert Proof System Instance 1 for Predicate Logic

Theorem
Let $\LL$ be the language of predicate logic.

Let $\mathscr H$ be instance 1 of a Hilbert proof system for predicate logic.

Then Proof by Contradiction is a derived rule of $\mathscr H$:

Proof
Suppose that $\Sigma, \phi \vdash_{\mathscr H} \bot$.

Note that $\bot \implies \neg\phi$ is a tautology.

Therefore, $\bot \implies \neg\phi$ is an axiom of $\mathscr H$, so that by Modus Ponendo Ponens:


 * $\Sigma, \phi \vdash_{\mathscr H} \neg \phi$

By Deduction Theorem for Hilbert Proof System for Predicate Logic, it follows that:


 * $\Sigma \vdash_{\mathscr H} \phi \implies \neg \phi$

Next, note that $\paren{ \phi \implies \neg \phi } \implies \neg \phi$ is a tautology and so an axiom of $\mathscr H$.

Hence by Modus Ponendo Ponens:


 * $\Sigma \vdash_{\mathscr H} \neg \phi$