Consider the following premises. For all x_5, if x_5 cooks, then x_5 is a doctor. For all x_5, if x_5 were a doctor, then x_5 is cooking. For all x_5, if x_5 were a doctor, then x_5 is cooking. Can we infer the following from them? There is at least one x_5 for which if is not true that x_5 will cook if and only if x_5 is a doctor.
 From the fact that for all x_5, if x_5 cooks, then x_5 is a doctor, and that for all x_5, if x_5 were a doctor, then x_5 is cooking we can infer that for all x_5, x_5 will cook if and only if x_5 is a doctor. From the fact that for all x_5, x_5 will cook if and only if x_5 is a doctor we can infer that for all x_5, if x_5 cooks, then x_5 is a doctor. Finally, from the fact that for all x_5, if x_5 cooks, then x_5 is a doctor, and that for all x_5, if x_5 were a doctor, then x_5 is cooking we can infer that for all x_5, x_5 will cook if and only if x_5 is a doctor, which contradicts that There is at least one x_5 for which if is not true that x_5 will cook if and only if x_5 is a doctor. Therefore, the answer is no.