1. A sequent with only atoms is valid iff some formula occurs on

both sides.

2. A, ¬ϕ ⇒ B iff A ⇒ B, ϕ

3. A ⇒ B, ¬ϕ iff A, ϕ ⇒ B

4. A, ϕ ∧ ψ ⇒ B iff A, ϕ, ψ ⇒ B

5. A ⇒ B, ϕ ∧ ψ iff both A ⇒ B, ϕ and A ⇒ B, ψ

These are facts about modal sequents...

How do we prove them?

For example 2. A, ¬ϕ ⇒ B iff A ⇒ B, ϕ

Where A and B = finite sets of wwfs

We suppose A, ¬ϕ ⇒ B first. Can we prove it without using contradiction method?

Supposing A not ⇒ B, ϕ and showing contradiction.