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