Extended Soundness 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 $$\mathbf H \vdash \mathbf A$$, 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 means No Model‎, $$\mathbf H \cup \left\{{\neg \mathbf A}\right\}$$ 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 logical consequence, $$\mathbf H \models \mathbf A$$.

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