Consider the following premises. For all x_5, if x_5 writes a letter, then x_5 is cooking. For all x_5, x_5 will not cook. For all x_5, x_5 will go running if and only if x_5 will write a letter. Can we infer the following from them? There is at least one x_5 for which x_5 will go running.
 From the fact that for all x_5, if x_5 writes a letter, then x_5 is cooking, and that for all x_5, x_5 will not cook we can infer that for all x_5, x_5 will not write a letter. From the fact that for all x_5, x_5 will go running if and only if x_5 will write a letter we can infer that for all x_5, if x_5 goes running, then x_5 is writing a letter. Finally, from the fact that for all x_5, if x_5 goes running, then x_5 is writing a letter, and that for all x_5, x_5 will not write a letter we can infer that for all x_5, x_5 will not go running, which contradicts that There is at least one x_5 for which x_5 will go running. Therefore, the answer is no.