Soundness Theorem for Hilbert Proof System

Theorem
Let $\mathscr H$ be instance 1 of a Hilbert proof system.

Let $\mathrm{BI}$ be the formal semantics of boolean interpretations.

Then $\mathscr H$ is a sound proof system for $\mathrm{BI}$:


 * Every $\mathscr H$-theorem is a tautology.

Proof
Recall the axioms of $\mathscr H$:

That these are tautologies is shown on, respectively:


 * True Statement is implied by Every Statement
 * Self-Distributive Law for Conditional


 * Rule of Transposition

That Modus Ponens infers tautologies from tautologies is shown on:


 * Modus Ponendo Ponens

Since:


 * All axioms of $\mathscr H$ are tautologies;
 * All rules of inference of $\mathscr H$ preserve tautologies

it is guaranteed that every formal proof in $\mathscr H$ results in a tautology.

That is, all $\mathscr H$-theorems are tautologies.