How do we know inference rules are correct?

Question

I know axioms are statements which are assumed to be true (meaning that axioms are not proved).

Theorems are statements which can be proved or has been proved. In the proofs of theorems we can use axioms, previously proven theorems and inference rules.

Inference rules is the part where I am a little bit confused. Do inference rules need to be proved or are they assumed to be true like axioms are? My understanding is inference rules are assumed to be true, but I am not sure at ALL!

If inference rules need to be proved then let's take for example Natural Deduction Calculus:

• Axioms: A\/~A; A=>A

How would I prove for example these inference rules:

1. Conjunction elimination: A /\B |- A,B
2. Implication elimination/Modus ponens: A, A=>B |- B

Answer 1

I take the example of modus ponens. You have to prove that $\forall A\forall B, A=>(A=>B)=>B$.