Consider the following premises. forall x: U(x) <-> W_3(x). forall x: ~U(x). forall x: Q(x) -> W_3(x). exists x: W(x) <-> W_3(x). forall x: U(x) -> Q(x). Can we infer exists x: ~W(x) from them?
 From forall x: U(x) <-> W_3(x) we can infer forall x: W_3(x) -> U(x). From forall x: U(x) -> Q(x), forall x: Q(x) -> W_3(x) we can infer forall x: U(x) -> W_3(x). From forall x: W_3(x) -> U(x), forall x: U(x) -> W_3(x) we can infer forall x: W_3(x) <-> U(x). From forall x: W_3(x) <-> U(x), forall x: ~U(x) we can infer forall x: ~W_3(x). Finally, from exists x: W(x) <-> W_3(x), forall x: ~W_3(x) we can infer exists x: ~W(x). Therefore, the answer is yes.