Consider the following premises. forall x: ~U_1(x). forall x: W_1(x) -> W_2(x). exists x: W_2(x) -> Q(x). forall x: Q(x) -> U_1(x). Can we infer exists x: ~W_1(x) from them?
 From forall x: Q(x) -> U_1(x), forall x: ~U_1(x) we can infer forall x: ~Q(x). From forall x: W_1(x) -> W_2(x), exists x: W_2(x) -> Q(x) we can infer exists x: W_1(x) -> Q(x). Finally, from exists x: W_1(x) -> Q(x), forall x: ~Q(x) we can infer exists x: ~W_1(x). Therefore, the answer is yes.