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Following A. Kuznetsov's outline, we restore Kuznetsov's syntactic proof ofthe assertoric equipollence of the intuitionistic propositional calculus andthe proof-intuitionistic calculus KM (Kuznetsov's Theorem). Then, we show thatthis property is true for a broad class of modal logics on an intuitionisticbasis, which includes, e.g., the modalized Heyting calculus mHC. The last factis one of two key properties necessary for the commutativity of a diagraminvolving the lattices of normal extensions of four well-known logics. Also, wegive an algebraic interpretation of the assertoric equipollence for subsystemsof KM.

On the Equipollence of the Calculi Int and KM