Consider the following premises. exists z: P_1(z). forall z: ~P_1(z) or P(z). forall z: ~U(z). Can we infer forall z: ~P(z) from them? If possible, name the inference rules being used at each step.
 From exists z: P_1(z) we can infer exists z: P_1(z) or U(z) via existential addition. From exists z: P_1(z) or U(z), forall z: ~P_1(z) or P(z) we can infer exists z: U(z) or P(z) via existential resolution. Finally, from exists z: U(z) or P(z), forall z: ~U(z) we can infer exists z: P(z) via existential disjunctive syllogism, which contradicts forall z: ~P(z). Therefore, the answer is no.