Consider the following premises. forall z: R_4(z) <-> P_3(z). forall z: ~R_4(z). forall z: P_3(z) -> R_4(z). Can we infer forall z: ~P_3(z) from them? If possible, name the inference rules being used at each step.
 From forall z: R_4(z) <-> P_3(z) we can infer forall z: R_4(z) -> P_3(z) via universal biconditional elimination. From forall z: R_4(z) -> P_3(z), forall z: P_3(z) -> R_4(z) we can infer forall z: R_4(z) <-> P_3(z) via universal biconditional introduction. Finally, from forall z: R_4(z) <-> P_3(z), forall z: ~R_4(z) we can infer forall z: ~P_3(z) via universal biconditional elimination. Therefore, the answer is yes.