Consider the following premises. w_1 -> w. w -> u. p_4 <-> u. Can we infer ~(w_1 -> p_4) from them? If possible, name the inference rules being used at each step.
 From p_4 <-> u we can infer u -> p_4 via biconditional elimination. From w -> u, u -> p_4 we can infer w -> p_4 via transitivity. Finally, from w_1 -> w, w -> p_4 we can infer w_1 -> p_4 via transitivity, which contradicts ~(w_1 -> p_4). Therefore, the answer is no.