Consider the following premises. p_3 -> p. s -> p_3. p -> s. p -> p_3. Can we infer ~(p <-> p_3) from them?
 From p -> s, s -> p_3 we can infer p -> p_3. From p_3 -> p, p -> p_3 we can infer p_3 <-> p. From p_3 <-> p we can infer p_3 -> p. Finally, from p -> p_3, p_3 -> p we can infer p <-> p_3, which contradicts ~(p <-> p_3). Therefore, the answer is no.