Consider the following premises. q <-> q_8. q_2 -> q. w -> q_2. Can we infer ~(w -> q_8) from them? If possible, name the inference rules being used at each step.
 From q <-> q_8 we can infer q -> q_8 via biconditional elimination. From q_2 -> q, q -> q_8 we can infer q_2 -> q_8 via transitivity. Finally, from w -> q_2, q_2 -> q_8 we can infer w -> q_8 via transitivity, which contradicts ~(w -> q_8). Therefore, the answer is no.