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