Consider the following premises. For all x_1, x_1 will write a letter if and only if x_1 is a server. There is at least one x_1 for which x_1 is a server if and only if x_1 will write a letter. Can we infer the following from them? For all x_1, if is not true that x_1 is a server if and only if x_1 will write a letter.
That contradicts the premises. Therefore the answer is no.