Consider the following premises. P_8(a_2) -> q_1. forall x_8: P_8(x_8) -> Q_8(x_8). forall x_8: ~Q_8(x_8). q_1 -> P_8(a_2). Can we infer q_1 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) we can infer forall x_8: ~P_8(x_8) via universal modus tollens. From forall x_8: ~P_8(x_8) we can infer ~P_8(a_2) via universal instantiation. From P_8(a_2) -> q_1, q_1 -> P_8(a_2) we can infer P_8(a_2) <-> q_1 via biconditional introduction. Finally, from P_8(a_2) <-> q_1, ~P_8(a_2) we can infer ~q_1 via biconditional elimination, which contradicts q_1. Therefore, the answer is no.