Consider the following premises. For all z_3, if z_3 were a cashier, then z_3 is writing a letter. For all z_3, if z_3 writes a letter, then z_3 is a cashier. For all z_3, if z_3 were a cashier, then z_3 is writing a letter. Can we infer the following from them? If David were a cashier, then David is writing a letter.
 From the fact that for all z_3, if z_3 were a cashier, then z_3 is writing a letter, and that for all z_3, if z_3 writes a letter, then z_3 is a cashier we can infer that for all z_3, z_3 is a cashier if and only if z_3 will write a letter. From the fact that for all z_3, z_3 is a cashier if and only if z_3 will write a letter we can infer that for all z_3, if z_3 writes a letter, then z_3 is a cashier. From the fact that for all z_3, if z_3 were a cashier, then z_3 is writing a letter, and that for all z_3, if z_3 writes a letter, then z_3 is a cashier we can infer that for all z_3, z_3 is a cashier if and only if z_3 will write a letter. From the fact that for all z_3, z_3 is a cashier if and only if z_3 will write a letter we can infer that David is a cashier if and only if David is writing a letter. Finally, from the fact that David is a cashier if and only if David is writing a letter we can infer that if David were a cashier, then David is writing a letter. Therefore, the answer is yes.