Consider the following premises. There is at least one y_1 for which if y_1 were a cashier, then y_1 is writing a letter. For all y_1, y_1 will not sleep. For all y_1, y_1 is a cashier if and only if y_1 will sleep. For all y_1, if y_1 writes a letter, then y_1 is a cashier. Can we infer the following from them? If we can, name the inference rule being used: For all y_1, y_1 will write a letter.
 From the fact that for all y_1, y_1 is a cashier if and only if y_1 will sleep, and that for all y_1, y_1 will not sleep we can infer that for all y_1, y_1 is not a cashier via universal biconditional elimination. From the fact that for all y_1, if y_1 writes a letter, then y_1 is a cashier, and that there is at least one y_1 for which if y_1 were a cashier, then y_1 is writing a letter we can infer that there is at least one y_1 for which y_1 will write a letter if and only if y_1 is a cashier via existential biconditional introduction. Finally, from the fact that there is at least one y_1 for which y_1 will write a letter if and only if y_1 is a cashier, and that for all y_1, y_1 is not a cashier we can infer that there is at least one y_1 for which y_1 will not write a letter via existential biconditional elimination, which contradicts that For all y_1, y_1 will write a letter. Therefore, the answer is no.