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 $$\mathbf H \models \mathbf A$$.