I have to prove that the following statement is valid.
If $φ$ and $ψ$ have no propositional letter in common, then $φ ∨ ψ$ is a tautology iff φ is a tautology or ψ is a tautology.
As this includes an if-and-only-if I have to prove two directions.
I started proving this by stating the following:
Let $φ$ be a contradiction and $φ ∨ ψ$ be a tautology. Because $φ$ is a contradiction, $φ ∨ ψ$ is equivalent to $ψ$. Therefore $ψ$ must be a tautology, because otherwise $φ ∨ ψ$ cannot be a tautology.
Am I on the right track? If so, how can I prove the other way? If not, can someone give me some advice on how to prove that?