Consider the following premises. forall x_8: Q_8(x_8) -> Q(x_8). exists x_8: R_2(x_8) <-> Q(x_8). forall x_8: R_2(x_8) -> Q_8(x_8). forall x_8: ~Q(x_8). Can we infer exists x_8: ~R_2(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: R_2(x_8) -> Q_8(x_8), forall x_8: Q_8(x_8) -> Q(x_8) we can infer forall x_8: R_2(x_8) -> Q(x_8) via universal transitivity. From exists x_8: R_2(x_8) <-> Q(x_8) we can infer exists x_8: Q(x_8) -> R_2(x_8) via existential biconditional elimination. From forall x_8: R_2(x_8) -> Q(x_8), exists x_8: Q(x_8) -> R_2(x_8) we can infer exists x_8: R_2(x_8) <-> Q(x_8) via existential biconditional introduction. Finally, from exists x_8: R_2(x_8) <-> Q(x_8), forall x_8: ~Q(x_8) we can infer exists x_8: ~R_2(x_8) via existential biconditional elimination. Therefore, the answer is yes.