Consider the following premises. forall x_8: P_8(x_8) -> Q_8(x_8). Q_8(a_7) -> ~q_2. forall x_8: Q_8(x_8) -> P_8(x_8). q_2. Can we infer P_8(a_7) 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 Q_8(a_7) -> ~q_2, q_2 we can infer ~Q_8(a_7) via modus tollens. Finally, from P_8(a_7) <-> Q_8(a_7), ~Q_8(a_7) we can infer ~P_8(a_7) via biconditional elimination, which contradicts P_8(a_7). Therefore, the answer is no.