Consider the following premises. forall z_3: P_2(z_3) -> P_3(z_3). forall z_3: P_2(z_3) or Q_2(z_3). exists z_3: Q_2(z_3) -> P_3(z_3). forall z_3: U_3(z_3). forall z_3: U_3(z_3) -> U(z_3). Can we infer forall z_3: ~(U(z_3) and P_3(z_3)) from them? If possible, name the inference rules being used at each step.
 From forall z_3: P_2(z_3) -> P_3(z_3), exists z_3: Q_2(z_3) -> P_3(z_3), forall z_3: P_2(z_3) or Q_2(z_3) we can infer exists z_3: P_3(z_3) via existential disjunction elimination. From forall z_3: U_3(z_3) -> U(z_3), forall z_3: U_3(z_3) we can infer forall z_3: U(z_3) via universal modus ponens. Finally, from forall z_3: U(z_3), exists z_3: P_3(z_3) we can infer exists z_3: U(z_3) and P_3(z_3) via existential conjunction, which contradicts forall z_3: ~(U(z_3) and P_3(z_3)). Therefore, the answer is no.