Consider the following premises. forall x: U(x). forall x: P_2(x) -> ~U(x). forall x: W(x) -> Q(x). exists x: Q(x) -> P_2(x). Can we infer exists x: ~W(x) from them?
 From forall x: P_2(x) -> ~U(x), forall x: U(x) we can infer forall x: ~P_2(x). From exists x: Q(x) -> P_2(x), forall x: ~P_2(x) we can infer exists x: ~Q(x). Finally, from forall x: W(x) -> Q(x), exists x: ~Q(x) we can infer exists x: ~W(x). Therefore, the answer is yes.