Compactness Theorem for Boolean Interpretations

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

Suppose every finite subset of $$\mathbf H$$ has a model.

Then $$\mathbf H$$ has a model.

Proof
Suppose $$\mathbf H$$ does not have a model.

By the Main Lemma of Propositional Calculus, $$\mathbf H$$ has a tableau confutation $$T$$.

Since each tableau confutation is a finite tableau, the set $$\mathbf H'$$ of all propositional WFFs in $$\mathbf H$$ used somewhere in $$T$$ is finite.

Now, let $$T'$$ be the labeled tree which is the same as $$T$$ but with root $$\mathbf H'$$ instead of $$\mathbf H$$.

Then $$T'$$ is a tableau confutation of $$\mathbf H'$$.

By the Extended Soundness Theorem of Propositional Calculus, $$\mathbf H'$$ has no models.

But this contradicts the assumption that all finite subsets of $$\mathbf H$$ have models.

Hence the result.