Consider the following premises. forall z_2: R_4(z_2) -> U(z_2). forall z_2: W(z_2) -> Q(z_2). forall z_2: Q(z_2) <-> P_1(z_2). forall z_2: Q_3(z_2) -> U(z_2). forall z_2: P_1(z_2) or W(z_2). forall z_2: R_4(z_2) or Q_3(z_2). Can we infer exists z_2: ~(Q(z_2) and U(z_2)) from them? If possible, name the inference rules being used at each step.
 From forall z_2: Q(z_2) <-> P_1(z_2) we can infer forall z_2: P_1(z_2) -> Q(z_2) via universal biconditional elimination. From forall z_2: R_4(z_2) -> U(z_2), forall z_2: Q_3(z_2) -> U(z_2), forall z_2: R_4(z_2) or Q_3(z_2) we can infer forall z_2: U(z_2) via universal disjunction elimination. From forall z_2: P_1(z_2) -> Q(z_2), forall z_2: W(z_2) -> Q(z_2), forall z_2: P_1(z_2) or W(z_2) we can infer forall z_2: Q(z_2) via universal disjunction elimination. Finally, from forall z_2: Q(z_2), forall z_2: U(z_2) we can infer forall z_2: Q(z_2) and U(z_2) via universal conjunction, which contradicts exists z_2: ~(Q(z_2) and U(z_2)). Therefore, the answer is no.