Consider the following premises. For all y_3, if y_3 writes a letter, then y_3 is cooking. For all y_3, if y_3 cooks, then y_3 is writing a letter. There is at least one y_3 for which y_3 will write a letter if and only if y_3 will play a game. For all y_3, y_3 will not play a game. Can we infer the following from them? For all y_3, y_3 will cook.
 From the fact that there is at least one y_3 for which y_3 will write a letter if and only if y_3 will play a game, and that for all y_3, y_3 will not play a game we can infer that there is at least one y_3 for which y_3 will not write a letter. From the fact that for all y_3, if y_3 cooks, then y_3 is writing a letter, and that for all y_3, if y_3 writes a letter, then y_3 is cooking we can infer that for all y_3, y_3 will cook if and only if y_3 will write a letter. Finally, from the fact that for all y_3, y_3 will cook if and only if y_3 will write a letter, and that there is at least one y_3 for which y_3 will not write a letter we can infer that there is at least one y_3 for which y_3 will not cook, which contradicts that For all y_3, y_3 will cook. Therefore, the answer is no.