Consider the following premises. ~Q_5(a_2). forall x_5: P_5(x_5) -> Q_5(x_5). forall x_5: P_5(x_5) or ~Q_5(x_5). Can we infer ~P_5(a_2) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_5(x_5) or ~Q_5(x_5) we can infer forall x_5: Q_5(x_5) -> P_5(x_5) via universal rewriting. From forall x_5: P_5(x_5) -> Q_5(x_5), forall x_5: Q_5(x_5) -> P_5(x_5) we can infer forall x_5: P_5(x_5) <-> Q_5(x_5) via universal biconditional introduction. From forall x_5: P_5(x_5) <-> Q_5(x_5) we can infer P_5(a_2) <-> Q_5(a_2) via universal instantiation. Finally, from P_5(a_2) <-> Q_5(a_2), ~Q_5(a_2) we can infer ~P_5(a_2) via biconditional elimination. Therefore, the answer is yes.