Completeness Theorem for Propositional Tableaus and Boolean Interpretations

Theorem
Tableau proofs (in terms of propositional tableaus) are a complete proof system for boolean interpretations.

That is, for every WFF $\mathbf A$:


 * $\models_{\mathrm{BI}} \mathbf A$ implies $\vdash_{\mathrm{PT}} \mathbf A$

Proof
This is a corollary of the Extended Completeness Theorem for Propositional Tableaus and Boolean Interpretations.

Namely, it is the special case $\mathbf H = \varnothing$.

Hence the result.

Also see
The Soundness Theorem for Propositional Tableaus and Boolean Interpretations in which it is proved that:
 * If $\vdash \mathbf A$ then $\models \mathbf A$.