Consider the following premises. exists x_1: R_4(x_1) <-> T(x_1). forall x_1: ~S_2(x_1). forall x_1: T(x_1) <-> R_4(x_1). forall x_1: T(x_1) -> S_2(x_1). Can we infer exists x_1: ~R_4(x_1) from them? If possible, name the inference rules being used at each step.
 From exists x_1: R_4(x_1) <-> T(x_1) we can infer exists x_1: T(x_1) -> R_4(x_1) via existential biconditional elimination. From forall x_1: T(x_1) <-> R_4(x_1) we can infer forall x_1: R_4(x_1) -> T(x_1) via universal biconditional elimination. From forall x_1: R_4(x_1) -> T(x_1), exists x_1: T(x_1) -> R_4(x_1) we can infer exists x_1: R_4(x_1) <-> T(x_1) via existential biconditional introduction. From forall x_1: T(x_1) -> S_2(x_1), forall x_1: ~S_2(x_1) we can infer forall x_1: ~T(x_1) via universal modus tollens. Finally, from exists x_1: R_4(x_1) <-> T(x_1), forall x_1: ~T(x_1) we can infer exists x_1: ~R_4(x_1) via existential biconditional elimination. Therefore, the answer is yes.