I'm trying to solve a problem which asks me to show that the meta-rule defined by deriving X(P) from X(a) preserves derivability (i.e. if ⊢X(a) then ⊢X(P) in modal system K, where a is a sentence letter which occurs zero or more times in an MPL-wff X(a), and X(P) is the result of replacing every instance of a with the wff P.
This would be relatively easy to do semantically and then use completeness of K to give the desired result, but the problem asks me to do this without making use of completeness.
In system K, I have the axioms
and the rules modus ponens and necessitation.
I really can't see how I can go from X(a) to X(P) using these, so I'd appreciate any help you could offer.
EDIT: Have replaced all greek letters with regular characters: note P, Q, R here represent arbitrary wffs and not just sentence letters