You ask how proof verifiers work, but it seems you are asking a somewhat more precise question: how do proof verifiers based on various higher-order logics work? If you are interested only in classical first-order logic (or less, e.g., propositional logic), then you don't need to worry about higher-order issues. (There are important caveats. For example, the expressiveness of plain first-order logic has limits that you might consider unwelcome for various purposes, such as its inability to capture, e.g., reachability in a graph.) The Mizar system, for example, is based on classical first-order logic plus (a rather strong) set theory.
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