Extended Soundness Theorem for Propositional Tableaus and Boolean Interpretations

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

Let $\mathbf A$ be a propositional formula.

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

Proof
We are given $\mathbf H$ and $\mathbf A$.

If $\mathbf H \vdash \mathbf A$, then there exists a tableau proof for it.

Thus we have a tableau confutation of $\mathbf H \cup \left\{{\neg \mathbf A}\right\}$.

So, by Tableau Confutation implies Unsatisfiable, $\mathbf H \cup \left\{{\neg \mathbf A}\right\}$ has no model.

That is, no model of $\mathbf H$ is also a model of $\neg \mathbf A$.

So if $\mathcal M$ is a model of $\mathbf H$, $\mathcal M$ is also a model of $\mathbf A$.

Thus, by definition of semantic consequence, $\mathbf H \models \mathbf A$.

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