Consider the following premises. forall z: R_2(z) <-> P_3(z). forall z: P(z). exists z: P_3(z). Can we infer forall z: ~R_2(z) from them? If possible, name the inference rules being used at each step.
 From exists z: P_3(z), forall z: P(z) we can infer exists z: P_3(z) and P(z) via existential conjunction. From forall z: R_2(z) <-> P_3(z) we can infer forall z: P_3(z) -> R_2(z) via universal biconditional elimination. From exists z: P_3(z) and P(z) we can infer exists z: P_3(z) via existential simplification. Finally, from forall z: P_3(z) -> R_2(z), exists z: P_3(z) we can infer exists z: R_2(z) via existential modus ponens, which contradicts forall z: ~R_2(z). Therefore, the answer is no.