Consider the following premises. forall z: ~U(z). forall z: P(z) -> U(z). forall z: U(z) -> P(z). Can we infer forall z: ~P(z) from them? If possible, name the inference rules being used at each step.
 From forall z: U(z) -> P(z), forall z: P(z) -> U(z) we can infer forall z: U(z) <-> P(z) via universal biconditional introduction. From forall z: U(z) <-> P(z) we can infer forall z: P(z) -> U(z) via universal biconditional elimination. Finally, from forall z: P(z) -> U(z), forall z: ~U(z) we can infer forall z: ~P(z) via universal modus tollens. Therefore, the answer is yes.