I am currently taking an introductory course to mathematical logic. We have started with propositional logic and today introduced the Gentzen style proof calculus. In order to prove that the soundness property of this we first needed the Principle of Rule Induction. My professor stated it as follows:
Theorem: Let P be a property closed under all basic rules of derivability. Also, let X be a set of sentences and $\alpha$ be a sentence. Then X $\vdash$ $\alpha$ implies P holds for the sequent (X,$\alpha$).
I understood the proof of this theorem, it was fairly simple proof by induction on the length of formulas. Where I am struggling is the intuition behind the application of the principle. Is it saying that if I can show that some property holds for all basic rules of the propositional calculus, then it will hold for all rules that are derivable in the propositional calculus? For example, in proving soundness, would I be showing that for all basic rules their syntactic provability implies they have semantic truth?