Consider the following premises. For all x_4, x_4 is poor if and only if x_4 is a soldier. There is at least one x_4 for which x_4 is a soldier if and only if x_4 will write a letter. For all x_4, x_4 is not poor. Can we infer the following from them? 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 poor if and only if x_4 is a soldier, and that for all x_4, x_4 is not poor we can infer that for all x_4, x_4 is not a soldier. Finally, from the fact that there is at least one x_4 for which x_4 is a soldier if and only if x_4 will write a letter, and that for all x_4, x_4 is not a soldier we can infer that there is at least one x_4 for which x_4 will not write a letter. Therefore, the answer is yes.