Consider the following premises. forall x: P(x). forall x_8: P(x_8) -> S_8(x_8). q_1 -> S_8(a_9). Can we infer ~S_8(a_9) from them? If possible, name the inference rules being used at each step.
 From forall x: P(x) we can infer P(a_9) via universal instantiation. From forall x_8: P(x_8) -> S_8(x_8) we can infer P(a_9) -> S_8(a_9) via universal instantiation. From P(a_9) we can infer P(a_9) or q_1 via addition. Finally, from P(a_9) -> S_8(a_9), q_1 -> S_8(a_9), P(a_9) or q_1 we can infer S_8(a_9) via disjunction elimination, which contradicts ~S_8(a_9). Therefore, the answer is no.