Completeness Theorem for Propositional Tableaus and Boolean Interpretations

Theorem
If a propositional formula is a tautology, then it has a tableau proof.

That is:
 * If $\models \mathbf A$ then $\vdash \mathbf A$.

Proof
This is a corollary of the Extended Completeness Theorem of Propositional Logic.

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

Hence the result.

Also see
The Soundness Theorem of Propositional Logic in which it is proved that:
 * If $\vdash \mathbf A$ then $\models \mathbf A$.