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