Consider the following premises. q_8 -> r_8. ~Q(i). Q(i) -> q_8. forall x: P(x) -> Q(x). r_8 -> P(i). Can we infer ~P(i) from them? If possible, name the inference rules being used at each step.
 From forall x: P(x) -> Q(x) we can infer P(i) -> Q(i) via universal instantiation. From Q(i) -> q_8, q_8 -> r_8 we can infer Q(i) -> r_8 via transitivity. From Q(i) -> r_8, r_8 -> P(i) we can infer Q(i) -> P(i) via transitivity. From P(i) -> Q(i), Q(i) -> P(i) we can infer P(i) <-> Q(i) via biconditional introduction. Finally, from P(i) <-> Q(i), ~Q(i) we can infer ~P(i) via biconditional elimination. Therefore, the answer is yes.