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