Consider the following premises. forall z: W_2(z) or W(z). forall z: U(z) or ~W(z). exists z: W_3(z) -> U(z). forall z: ~W_2(z) or W_3(z). Can we infer forall z: ~U(z) from them?
 From forall z: W_2(z) or W(z), forall z: ~W_2(z) or W_3(z) we can infer forall z: W(z) or W_3(z). From forall z: U(z) or ~W(z) we can infer forall z: W(z) -> U(z). Finally, from forall z: W(z) -> U(z), exists z: W_3(z) -> U(z), forall z: W(z) or W_3(z) we can infer exists z: U(z), which contradicts forall z: ~U(z). Therefore, the answer is no.