Provable by Gentzen Proof System iff Negation has Closed Tableau

Theorem

Let $\mathscr G$ be instance $1$ of a Gentzen proof system.

Formula

Let $\mathbf A$ be a WFF of propositional logic.

Then $\mathbf A$ is a $\mathscr G$-theorem if and only if:

$\neg \mathbf A$ has a closed semantic tableau

where $\neg \mathbf A$ is the negation of $\mathbf A$.

Set of Formulas

Let $U$ be a set of WFFs of propositional logic.

Then:

$U$ is a $\mathscr G$-theorem

if and only if:

$\bar U$ has a closed semantic tableau

where $\bar U$ is the set comprising the logical complements of all WFFs in $U$.