Consider the following premises. For all x_11, x_11 will cook if and only if x_11 will play a game. There is at least one x_11 for which x_11 will sleep if and only if x_11 will play a game. For all x_11, x_11 will not sleep. Can we infer the following from them? There is at least one x_11 for which x_11 will not cook.
 From the fact that for all x_11, x_11 will cook if and only if x_11 will play a game we can infer that for all x_11, if x_11 cooks, then x_11 is playing a game. From the fact that there is at least one x_11 for which x_11 will sleep if and only if x_11 will play a game, and that for all x_11, x_11 will not sleep we can infer that there is at least one x_11 for which x_11 will not play a game. Finally, from the fact that for all x_11, if x_11 cooks, then x_11 is playing a game, and that there is at least one x_11 for which x_11 will not play a game we can infer that there is at least one x_11 for which x_11 will not cook. Therefore, the answer is yes.