Consider the following premises. forall z_2: R_1(z_2) <-> S(z_2). exists z_2: S(z_2) -> R_1(z_2). forall z_2: ~S(z_2). Can we infer exists z_2: ~R_1(z_2) from them? If possible, name the inference rules being used at each step.
 From forall z_2: R_1(z_2) <-> S(z_2) we can infer forall z_2: R_1(z_2) -> S(z_2) via universal biconditional elimination. From exists z_2: S(z_2) -> R_1(z_2), forall z_2: R_1(z_2) -> S(z_2) we can infer exists z_2: S(z_2) <-> R_1(z_2) via existential biconditional introduction. Finally, from exists z_2: S(z_2) <-> R_1(z_2), forall z_2: ~S(z_2) we can infer exists z_2: ~R_1(z_2) via existential biconditional elimination. Therefore, the answer is yes.