Semantic Tableau Algorithm is Decision Procedure for Tautologies

Theorem
The Semantic Tableau Algorithm is a decision procedure for tautologies.

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

The Semantic Tableau Algorithm applied to $\neg \mathbf A$ yields a completed tableau for $\neg \mathbf A$.

By Corollary 2 to Soundness and Completeness of Semantic Tableaus, this completed tableau decides if $\mathbf A$ is a tautology.