Consider the following premises. w_1 -> w. p_6 -> w_1. w <-> p_6. Can we infer ~(p_6 <-> w) from them?
 From p_6 -> w_1, w_1 -> w we can infer p_6 -> w. From w <-> p_6 we can infer w -> p_6. Finally, from p_6 -> w, w -> p_6 we can infer p_6 <-> w, which contradicts ~(p_6 <-> w). Therefore, the answer is no.