Consider the following premises. There is at least one y_4 for which y_4 is an astronaut or y_4 is not bored. For all y_4, y_4 is an astronaut if and only if y_4 will cook. For all y_4, y_4 will cook or y_4 is bored. Can we infer the following from them? There is at least one y_4 for which y_4 is an astronaut.
 From the fact that there is at least one y_4 for which y_4 is an astronaut or y_4 is not bored we can infer that there is at least one y_4 for which if y_4 were bored, then y_4 is an astronaut. From the fact that for all y_4, y_4 is an astronaut if and only if y_4 will cook we can infer that for all y_4, if y_4 cooks, then y_4 is an astronaut. Finally, from the fact that for all y_4, if y_4 cooks, then y_4 is an astronaut, that there is at least one y_4 for which if y_4 were bored, then y_4 is an astronaut, and that for all y_4, y_4 will cook or y_4 is bored we can infer that there is at least one y_4 for which y_4 is an astronaut. Therefore, the answer is yes.