Consider the following premises. For all x_4, if x_4 writes a letter, then x_4 is a soldier. For all x_4, x_4 will not cook. There is at least one x_4 for which if x_4 were a soldier, then x_4 is cooking. For all x_4, if x_4 cooks, then x_4 is writing a letter. 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, if x_4 writes a letter, then x_4 is a soldier, and that there is at least one x_4 for which if x_4 were a soldier, then x_4 is cooking we can infer that there is at least one x_4 for which if x_4 writes a letter, then x_4 is cooking via existential transitivity. From the fact that for all x_4, if x_4 cooks, then x_4 is writing a letter, and that there is at least one x_4 for which if x_4 writes a letter, then x_4 is cooking we can infer that there is at least one x_4 for which x_4 will cook if and only if x_4 will write a letter via existential biconditional introduction. Finally, from the fact that there is at least one x_4 for which x_4 will cook if and only if x_4 will write a letter, and that for all x_4, x_4 will not cook we can infer that there is at least one x_4 for which x_4 will not write a letter via existential biconditional elimination. Therefore, the answer is yes.