Levin's and Prucnal's theorems on Medvedev's logic of finite problems

The purpose of this note is to provide a transparent and unified retelling of both Skvortsov’s proof of the structural completeness of Medvedev’s logic of finite problems, which is a classical result originally due to Prucnal, and of Levin’s proof that Medvedev’s logic of finite problems is the largest extension of the (weak) Kreisel-Putnam logic with the disjunction property. Presenting both results together allows us to simplify their presentation, as they both hinge on the same lemma. There is no novel content in this note, its purpose is merely to present the material in a more accessible way.