Completeness Theorem for Hilbert Proof System Instance 2 and Boolean Interpretations

Theorem
Instance 2 of the Hilbert proof systems is a complete proof system for boolean interpretations.

That is, for every WFF $\mathbf A$:


 * $\models_{\mathrm{BI}} \mathbf A$ implies $\vdash_{\mathscr H_2} \mathbf A$

Also see
The Soundness Theorem for Hilbert Proof System Instance 2 and Boolean Interpretations in which it is proved that:
 * If $\vdash \mathbf A$ then $\models \mathbf A$.