Consider the following premises. s -> s_2. s <-> s_2. ~s_2. Can we infer ~s or p_4 from them?
 From s <-> s_2 we can infer s_2 -> s. From s_2 -> s, s -> s_2 we can infer s_2 <-> s. From s_2 <-> s, ~s_2 we can infer ~s. Finally, from ~s we can infer ~s or p_4. Therefore, the answer is yes.