Soundness Theorem for Propositional Tableaus and Boolean Interpretations

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

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

Proof
This is a corollary of the Extended Soundness Theorem of Propositional Calculus:

Let $$\mathbf H$$ be a countable set of logical formulas.

Let $$\mathbf A$$ be a logical formula.

If $$\mathbf H \vdash \mathbf A$$, then $$\mathbf H \models \mathbf A$$.

In this case, we have $$\mathbf H = \varnothing$$.

Hence the result.

Also see
The Completeness Theorem of Propositional Calculus in which is proved:
 * If $$\models \mathbf A$$ then $$\vdash \mathbf A$$.