Soundness Theorem for Hilbert Proof System for Predicate Logic

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

Let $\mathrm{PL}$ be the formal semantics of structures for predicate logic.

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


 * Every $\mathscr H$-provable consequence is a $\mathrm{PL}$-semantic consequence.

Proof
By Axioms of Hilbert Proof System Instance 1 for Predicate Logic are Tautologies, the axioms of $\mathscr H$ are tautologies.

By Modus Ponendo Ponens for Semantic Consequence in Predicate Logic, the sole rule of inference of $\mathscr H$ preserves $\mathrm{PL}$-semantic consequence.