Consider the following premises. forall z_1: P(z_1) -> T(z_1). exists z_1: R_4(z_1) or ~P_3(z_1). forall z_1: P_3(z_1) -> P(z_1). forall z_1: T(z_1) -> P_3(z_1). Can we infer exists z_1: P(z_1) -> R_4(z_1) from them? If possible, name the inference rules being used at each step.
 From exists z_1: R_4(z_1) or ~P_3(z_1) we can infer exists z_1: P_3(z_1) -> R_4(z_1) via existential rewriting. From forall z_1: P(z_1) -> T(z_1), forall z_1: T(z_1) -> P_3(z_1) we can infer forall z_1: P(z_1) -> P_3(z_1) via universal transitivity. From forall z_1: P(z_1) -> P_3(z_1), forall z_1: P_3(z_1) -> P(z_1) we can infer forall z_1: P(z_1) <-> P_3(z_1) via universal biconditional introduction. From forall z_1: P(z_1) <-> P_3(z_1) we can infer forall z_1: P(z_1) -> P_3(z_1) via universal biconditional elimination. Finally, from forall z_1: P(z_1) -> P_3(z_1), exists z_1: P_3(z_1) -> R_4(z_1) we can infer exists z_1: P(z_1) -> R_4(z_1) via existential transitivity. Therefore, the answer is yes.