Consider the following premises. forall x: W_1(x) or ~Q_2(x). forall x: W_1(x) -> Q_2(x). forall x: ~W_1(x). Can we infer forall x: ~Q_2(x) from them?
 From forall x: W_1(x) or ~Q_2(x) we can infer forall x: Q_2(x) -> W_1(x). From forall x: Q_2(x) -> W_1(x), forall x: W_1(x) -> Q_2(x) we can infer forall x: Q_2(x) <-> W_1(x). Finally, from forall x: Q_2(x) <-> W_1(x), forall x: ~W_1(x) we can infer forall x: ~Q_2(x). Therefore, the answer is yes.