Consider the following premises. p_1 -> u. u <-> r. r -> p_1. Can we infer ~(r <-> p_1) from them? If possible, name the inference rules being used at each step.
 From u <-> r we can infer u -> r via biconditional elimination. From p_1 -> u, u -> r we can infer p_1 -> r via transitivity. Finally, from r -> p_1, p_1 -> r we can infer r <-> p_1 via biconditional introduction, which contradicts ~(r <-> p_1). Therefore, the answer is no.