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