Consider the following premises. forall z_1: P_2(z_1) -> R_4(z_1). forall z_1: R_4(z_1) -> P_2(z_1). forall z_1: ~R_4(z_1). forall z_1: T(z_1) -> P_2(z_1). Can we infer exists z_1: T(z_1) from them? If possible, name the inference rules being used at each step.
 From forall z_1: R_4(z_1) -> P_2(z_1), forall z_1: P_2(z_1) -> R_4(z_1) we can infer forall z_1: R_4(z_1) <-> P_2(z_1) via universal biconditional introduction. From forall z_1: R_4(z_1) <-> P_2(z_1), forall z_1: ~R_4(z_1) we can infer forall z_1: ~P_2(z_1) via universal biconditional elimination. Finally, from forall z_1: T(z_1) -> P_2(z_1), forall z_1: ~P_2(z_1) we can infer forall z_1: ~T(z_1) via universal modus tollens, which contradicts exists z_1: T(z_1). Therefore, the answer is no.