Consider the following premises. Barbara is not rich. For all y_2, if y_2 were rich, then y_2 is cooking. For all y_2, y_2 will cook if and only if y_2 is rich. Can we infer the following from them? Barbara is not cooking.
 From the fact that for all y_2, y_2 will cook if and only if y_2 is rich we can infer that for all y_2, if y_2 cooks, then y_2 is rich. From the fact that for all y_2, if y_2 cooks, then y_2 is rich, and that for all y_2, if y_2 were rich, then y_2 is cooking we can infer that for all y_2, y_2 will cook if and only if y_2 is rich. From the fact that for all y_2, y_2 will cook if and only if y_2 is rich we can infer that Barbara is cooking if and only if Barbara is rich. Finally, from the fact that Barbara is cooking if and only if Barbara is rich, and that Barbara is not rich we can infer that Barbara is not cooking. Therefore, the answer is yes.