Consider the following premises. forall z: P(z) <-> R_4(z). forall z: P(z) -> P_4(z). exists z: P_4(z) -> P(z). forall z: ~P_4(z). Can we infer exists z: ~R_4(z) from them? If possible, name the inference rules being used at each step.
 From forall z: P(z) -> P_4(z), exists z: P_4(z) -> P(z) we can infer exists z: P(z) <-> P_4(z) via existential biconditional introduction. From exists z: P(z) <-> P_4(z), forall z: ~P_4(z) we can infer exists z: ~P(z) via existential biconditional elimination. Finally, from forall z: P(z) <-> R_4(z), exists z: ~P(z) we can infer exists z: ~R_4(z) via existential biconditional elimination. Therefore, the answer is yes.