Consider the following premises. w -> r. p_9 <-> r. p_2 -> w. Can we infer ~(p_2 -> p_9) from them? If possible, name the inference rules being used at each step.
 From p_9 <-> r we can infer r -> p_9 via biconditional elimination. From p_2 -> w, w -> r we can infer p_2 -> r via transitivity. Finally, from p_2 -> r, r -> p_9 we can infer p_2 -> p_9 via transitivity, which contradicts ~(p_2 -> p_9). Therefore, the answer is no.