Consider the following premises. For all z_1, z_1 will not write a letter. For all z_1, if z_1 were a server, then z_1 is writing a letter. For all z_1, z_1 is a server if and only if z_1 will work. There is at least one z_1 for which if z_1 writes a letter, then z_1 is working. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z_1 for which z_1 is not a server.
 From the fact that for all z_1, z_1 is a server if and only if z_1 will work we can infer that for all z_1, if z_1 works, then z_1 is a server via universal biconditional elimination. From the fact that there is at least one z_1 for which if z_1 writes a letter, then z_1 is working, and that for all z_1, if z_1 works, then z_1 is a server we can infer that there is at least one z_1 for which if z_1 writes a letter, then z_1 is a server via existential transitivity. From the fact that for all z_1, if z_1 were a server, then z_1 is writing a letter, and that there is at least one z_1 for which if z_1 writes a letter, then z_1 is a server we can infer that there is at least one z_1 for which z_1 is a server if and only if z_1 will write a letter via existential biconditional introduction. Finally, from the fact that there is at least one z_1 for which z_1 is a server if and only if z_1 will write a letter, and that for all z_1, z_1 will not write a letter we can infer that there is at least one z_1 for which z_1 is not a server via existential biconditional elimination. Therefore, the answer is yes.