Consider the following premises. There is at least one z_2 for which z_2 will write a letter if and only if z_2 will sleep. For all z_2, z_2 will write a letter if and only if z_2 will sleep. Can we infer the following from them? For all z_2, if is not true that z_2 will write a letter if and only if z_2 will sleep.
That contradicts the premises. Therefore the answer is no.