Consider the following premises. There is at least one x_6 for which if x_6 cooks, then x_6 is writing a letter. For all x_6, x_6 will not write a letter. For all x_6, if x_6 writes a letter, then x_6 is cooking. Can we infer the following from them? For all x_6, x_6 will cook.
 From the fact that there is at least one x_6 for which if x_6 cooks, then x_6 is writing a letter, and that for all x_6, if x_6 writes a letter, then x_6 is cooking we can infer that there is at least one x_6 for which x_6 will cook if and only if x_6 will write a letter. From the fact that there is at least one x_6 for which x_6 will cook if and only if x_6 will write a letter we can infer that there is at least one x_6 for which if x_6 cooks, then x_6 is writing a letter. Finally, from the fact that there is at least one x_6 for which if x_6 cooks, then x_6 is writing a letter, and that for all x_6, x_6 will not write a letter we can infer that there is at least one x_6 for which x_6 will not cook, which contradicts that For all x_6, x_6 will cook. Therefore, the answer is no.