Derivation of implicational propositional axioms

Question

Is there a way to subtract and add properties of axioms to generate new axioms?

For example:

{L} = {P S K} // natural deduction

{P S K} = {P H K I} // natural deduction

{S K} = {?} // constructive logic

{K I} = {?}

Where:

L = ((A -> B) -> C) -> ((C -> A) -> (D -> A)) // Łukasiewicz's axiom system

P = ((A -> B) -> A) -> A // Piece's Law

H = (A -> B) -> ((B -> C) -> (A -> C)) // Weak hypothetical syllogism

S = (A -> (B -> C)) -> ((A -> B) -> (A -> C))

K = A -> (B -> A)

I = A -> A

I want to be able to be able to add and subtract axioms such that P + S + K = P + H + K + I implies S encodes the properties of H + I

I'm probably using unjustified assumptions here. For example, I assume you can derive S from H and I, without using P or K. Ideally, there would be a way to automate the process of constructing and destructing axioms (though it'd probably be NP hard).