Usually, when deriving algebraizability conditions for propositional Hilbert-style deductive systems, semantical considerations are used explicitly. By formalizing the metalogics ot these proofs it is possible to restrict them onto the formula algebra only. The advantages of this approach include more structured proofs, correlating with the ones for Gentzen-style systems, and syntactical clarifications and ramifications of previously known results. To do this, for a given Hilbert-style deductive system $S$, we define an abstract Gentzen-style system which formalizes some aspects of metalogical reasonings about $S$, and then, by using this Gentzen system, derive the protoalgebraicity condition.