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 $$\vdash \mathbf{A}$$.