Consider the following premises. Q_13(a_12) -> R_10(a_10). forall x_10: ~R_10(x_10). forall x_13: P_13(x_13) -> Q_13(x_13). forall x_13: Q_13(x_13) -> P_13(x_13). Can we infer ~P_13(a_12) from them? If possible, name the inference rules being used at each step.
 From forall x_13: P_13(x_13) -> Q_13(x_13), forall x_13: Q_13(x_13) -> P_13(x_13) we can infer forall x_13: P_13(x_13) <-> Q_13(x_13) via universal biconditional introduction. From forall x_13: P_13(x_13) <-> Q_13(x_13) we can infer P_13(a_12) <-> Q_13(a_12) via universal instantiation. From forall x_10: ~R_10(x_10) we can infer ~R_10(a_10) via universal instantiation. From Q_13(a_12) -> R_10(a_10), ~R_10(a_10) we can infer ~Q_13(a_12) via modus tollens. Finally, from P_13(a_12) <-> Q_13(a_12), ~Q_13(a_12) we can infer ~P_13(a_12) via biconditional elimination. Therefore, the answer is yes.