## Not sure if my logical equivalences are correct here or not

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I have a problem here that I'm trying to solve. The question asks me to check whether the argument is valid or not without using a truth table. I used equivalence laws to reduce the statement down to B ∨ Q. My conclusion now is that this argument is not valid because B ∨ Q isn't always true itself. My questions were, Is this a proper way to prove the validity? and, are my reductions correct from the equivalence laws? Thank you very much.

((B → ¬Q) ∧ ¬B) → Q ⇔ ((¬B ∨ ¬Q) ∧ ¬B) → Q     (conditional law)
⇔ ¬((¬B ∨ ¬Q) ∧ ¬B) ∨ Q    (conditional law)
⇔ (¬ (¬B ∨ ¬Q) ∨ B) ∨ Q    (DeMorgan’s law)
⇔ ((B ∧ Q) ∨ B) ∨ Q        (DeMorgan’s law)
⇔ (B ∨ (B ∧ Q)) ∨ Q        (commutative law)
⇔ B ∨ Q                    (absorption law)


Are you allowed to use truth trees? It seems like an easier approach for this argument. If you complete a truth tree (true premises, negated conclusion), you will have open branches, which means you were correct - the argument indeed fails when both B and Q are false.