Consider the following premises. forall z: Q(z) -> R_1(z). forall z: R_1(z) -> W(z). forall z: W(z) -> Q(z). Can we infer exists z: ~(W(z) -> Q(z)) from them? If possible, name the inference rules being used at each step.
 From forall z: Q(z) -> R_1(z), forall z: R_1(z) -> W(z) we can infer forall z: Q(z) -> W(z) via universal transitivity. From forall z: W(z) -> Q(z), forall z: Q(z) -> W(z) we can infer forall z: W(z) <-> Q(z) via universal biconditional introduction. Finally, from forall z: W(z) <-> Q(z) we can infer forall z: W(z) -> Q(z) via universal biconditional elimination, which contradicts exists z: ~(W(z) -> Q(z)). Therefore, the answer is no.