Consider the following premises. forall z: T(z) -> P(z). forall z: W_3(z) or W_4(z). forall z: W_4(z) -> P(z). forall z: W_3(z) -> T(z). Can we infer exists z: ~P(z) from them? If possible, name the inference rules being used at each step.
 From forall z: W_3(z) -> T(z), forall z: T(z) -> P(z) we can infer forall z: W_3(z) -> P(z) via universal transitivity. Finally, from forall z: W_3(z) -> P(z), forall z: W_4(z) -> P(z), forall z: W_3(z) or W_4(z) we can infer forall z: P(z) via universal disjunction elimination, which contradicts exists z: ~P(z). Therefore, the answer is no.