Soundness and Completeness of Gentzen Proof System

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

Let $\mathrm{BI}$ be the formal semantics of boolean interpretations.

Then $\mathscr G$ is a sound and complete proof system for $\mathrm{BI}$.

Proof
This is an immediate consequence of:


 * Provable by Gentzen Proof System iff Negation has Closed Tableau
 * Soundness and Completeness of Semantic Tableaus