Consider the following premises. forall x_9: P_9(x_9) <-> Q(x_9). q or Q(a_5). q -> P_9(a_5). Can we infer ~P_9(a_5) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) <-> Q(x_9) we can infer forall x_9: Q(x_9) -> P_9(x_9) via universal biconditional elimination. From forall x_9: Q(x_9) -> P_9(x_9) we can infer Q(a_5) -> P_9(a_5) via universal instantiation. Finally, from q -> P_9(a_5), Q(a_5) -> P_9(a_5), q or Q(a_5) we can infer P_9(a_5) via disjunction elimination, which contradicts ~P_9(a_5). Therefore, the answer is no.