Consider the following premises. forall x_8: Q_8(x_8) -> P_8(x_8). forall x_8: P_8(x_8) -> Q_8(x_8). Q_8(a_7) -> r. Can we infer P_8(a_7) -> r from them? If possible, name the inference rules being used at each step.
 From forall x_8: P_8(x_8) -> Q_8(x_8), forall x_8: Q_8(x_8) -> P_8(x_8) we can infer forall x_8: P_8(x_8) <-> Q_8(x_8) via universal biconditional introduction. From forall x_8: P_8(x_8) <-> Q_8(x_8) we can infer P_8(a_7) <-> Q_8(a_7) via universal instantiation. From P_8(a_7) <-> Q_8(a_7) we can infer P_8(a_7) -> Q_8(a_7) via biconditional elimination. Finally, from P_8(a_7) -> Q_8(a_7), Q_8(a_7) -> r we can infer P_8(a_7) -> r via transitivity. Therefore, the answer is yes.