Consider the following premises. forall z_3: U_2(z_3) <-> P_1(z_3). forall z_3: U_2(z_3) -> P_1(z_3). Can we infer exists z_3: ~(P_1(z_3) <-> U_2(z_3)) from them? If possible, name the inference rules being used at each step.
 From forall z_3: U_2(z_3) <-> P_1(z_3) we can infer forall z_3: P_1(z_3) -> U_2(z_3) via universal biconditional elimination. Finally, from forall z_3: P_1(z_3) -> U_2(z_3), forall z_3: U_2(z_3) -> P_1(z_3) we can infer forall z_3: P_1(z_3) <-> U_2(z_3) via universal biconditional introduction, which contradicts exists z_3: ~(P_1(z_3) <-> U_2(z_3)). Therefore, the answer is no.