Consider the following premises. forall x: P_2(x) -> Q(x). p_1 -> Q(a). P_2(a) or p_1. Can we infer ~Q(a) from them?
 From forall x: P_2(x) -> Q(x) we can infer P_2(a) -> Q(a). Finally, from P_2(a) -> Q(a), p_1 -> Q(a), P_2(a) or p_1 we can infer Q(a), which contradicts ~Q(a). Therefore, the answer is no.