Consider the following premises. There is at least one x_12 for which x_12 is a server if and only if x_12 will work. For all x_12, x_12 will play a game if and only if x_12 is a server. For all x_12, if x_12 works, then x_12 is playing a game. Can we infer the following from them? For all x_12, if is not true that x_12 is a server if and only if x_12 will work.
That contradicts the premises. Therefore the answer is no.