Extended Completeness Theorem for Propositional Tableaus and Boolean Interpretations

Theorem
Let $$\mathbf{H}$$ be a countable 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 $$\mathbf{H} \models \mathbf{A}$$.