Consider the following premises. forall x: R_4(x) -> P(x). forall x: R_4(x) -> P(x). ~P(b_4). forall x: P(x) -> R_4(x). Can we infer ~R_4(b_4) from them? If possible, name the inference rules being used at each step.
 From forall x: P(x) -> R_4(x), forall x: R_4(x) -> P(x) we can infer forall x: P(x) <-> R_4(x) via universal biconditional introduction. From forall x: P(x) <-> R_4(x) we can infer forall x: P(x) -> R_4(x) via universal biconditional elimination. From forall x: R_4(x) -> P(x), forall x: P(x) -> R_4(x) we can infer forall x: R_4(x) <-> P(x) via universal biconditional introduction. From forall x: R_4(x) <-> P(x) we can infer R_4(b_4) <-> P(b_4) via universal instantiation. Finally, from R_4(b_4) <-> P(b_4), ~P(b_4) we can infer ~R_4(b_4) via biconditional elimination. Therefore, the answer is yes.