Extended Completeness Theorem for Propositional Tableaus and Boolean Interpretations

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

Let $\mathbf A$ be a logical formula.

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

Proof
Suppose $\mathbf A$ is a semantic consequence of $\mathbf H$.

Then $\mathbf H \cup \left\{{\neg \mathbf A}\right\}$ has no models.

By the Finite Main Lemma, this set has a tableau confutation, which is a tableau proof of $\mathbf A$ from $\mathbf H$.

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