The methodology is the best you can do; proving anything is basically establishing that two descriptions are equivalent.
With formal systems you usually have one definition that written to make it obviously correct and another that is more "pragmatic". In the case of propositional logic the obviously correct formalization is truth tables which are completely untractable to work with for large numbers of variables but are very simple conceptually. The "pragmatic" formalization is the logical connectives like and, or, implication etc that we normally think of as propositional logic. When they prove that propositional logic is sound they mean that all propositions give the same result as a truth table when evaluated and when they prove it is complete they mean that all truth tables give the same result as a proposition and thus propositional logic and truth tables both formalize the same concept.
There is no concept of "correct" in formal systems because it is inherently an informal concept meaning it does what it is supposed to do.