Consider the following premises. forall x_8: R_8(x_8) -> Q_4(x_8). ~R_8(a_2). forall x_8: Q_8(x_8) -> R_8(x_8). forall x_8: R(x_8) -> Q_8(x_8). forall x_8: Q_4(x_8) -> R(x_8). Can we infer ~R(a_2) from them? If possible, name the inference rules being used at each step.
 From forall x_8: R(x_8) -> Q_8(x_8), forall x_8: Q_8(x_8) -> R_8(x_8) we can infer forall x_8: R(x_8) -> R_8(x_8) via universal transitivity. From forall x_8: R_8(x_8) -> Q_4(x_8), forall x_8: Q_4(x_8) -> R(x_8) we can infer forall x_8: R_8(x_8) -> R(x_8) via universal transitivity. From forall x_8: R(x_8) -> R_8(x_8), forall x_8: R_8(x_8) -> R(x_8) we can infer forall x_8: R(x_8) <-> R_8(x_8) via universal biconditional introduction. From forall x_8: R(x_8) <-> R_8(x_8) we can infer R(a_2) <-> R_8(a_2) via universal instantiation. Finally, from R(a_2) <-> R_8(a_2), ~R_8(a_2) we can infer ~R(a_2) via biconditional elimination. Therefore, the answer is yes.