Infinitely many functional pre-complete classes of formulas in the propositional provability intuitionistic logic

We consider the propositional provability intuitionistic logic $I^\Delta$, introduced by A. V. Kuznetsov [2]. We prove that there are infinitely many classes of formulas in the calculus of $I^\Delta$, which are pre-complete with respect to functional expressibility in $I^\Delta$. This result is stronger than an ealier one stated by the author in [1].