Consider the following premises. forall z: U(z) -> P_1(z). forall z: P_1(z) -> P(z). exists z: P(z) -> U(z). forall z: ~P(z). forall z: P_1(z) -> P(z). Can we infer exists z: ~P_1(z) from them? If possible, name the inference rules being used at each step.
 From exists z: P(z) -> U(z), forall z: U(z) -> P_1(z) we can infer exists z: P(z) -> P_1(z) via existential transitivity. From exists z: P(z) -> P_1(z), forall z: P_1(z) -> P(z) we can infer exists z: P(z) <-> P_1(z) via existential biconditional introduction. From exists z: P(z) <-> P_1(z) we can infer exists z: P(z) -> P_1(z) via existential biconditional elimination. From forall z: P_1(z) -> P(z), exists z: P(z) -> P_1(z) we can infer exists z: P_1(z) <-> P(z) via existential biconditional introduction. Finally, from exists z: P_1(z) <-> P(z), forall z: ~P(z) we can infer exists z: ~P_1(z) via existential biconditional elimination. Therefore, the answer is yes.