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, $$\mathbf{H} \models \mathbf{A}$$.