Hi! Could anyone tell me how you prove the wfff (well-formed formula) (◊p V ◊q) → ◊(p V q) in the minimal modal propositional logic system called NK? You can only use two rules (the Necessitation Rule (If A is a theorem of K, then so is □A) and the Modus Ponens inference rule (If P, then Q. - P. - Therefore, Q.)) and one axiom (Distribution Axiom: □(A→B) → (□A→□B)).
I've got this far:
1. (p V q) → (p V q) [pL]
2. □((p V q) → (p V q)) [1, N]
3. □ (p V q) → □(p V q) [2, K]
4. (□p V □q) → □(p V q) [3, K]
5. (¬◊¬p v ¬◊¬q) → ¬◊¬(p V q) [4, def □]
6. (◊p V ◊q) → ◊(p V q) [pL]
I thought this was the correct proof, but I could be wrong.
You can find more information on modal logic on the following links:
Thanks in advance!