Consider the following premises. If Michael were a cashier, then William is writing a letter. If William writes a letter, then Michael is a cashier. For all y_4, y_4 is not a cashier. Can we infer the following from them? William is not writing a letter.
 From the fact that for all y_4, y_4 is not a cashier we can infer that Michael is not a cashier. From the fact that if Michael were a cashier, then William is writing a letter, and that if William writes a letter, then Michael is a cashier we can infer that Michael is a cashier if and only if William is writing a letter. Finally, from the fact that Michael is a cashier if and only if William is writing a letter, and that Michael is not a cashier we can infer that William is not writing a letter. Therefore, the answer is yes.