Consider the following premises. For all x_4, x_4 is a cashier if and only if x_4 will write a letter. There is at least one x_4 for which x_4 is not a cashier. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_4 for which x_4 will not write a letter.
 From the fact that for all x_4, x_4 is a cashier if and only if x_4 will write a letter we can infer that for all x_4, if x_4 writes a letter, then x_4 is a cashier via universal biconditional elimination. Finally, from the fact that for all x_4, if x_4 writes a letter, then x_4 is a cashier, and that there is at least one x_4 for which x_4 is not a cashier we can infer that there is at least one x_4 for which x_4 will not write a letter via existential modus tollens. Therefore, the answer is yes.