Consider the following premises. For all x_5, x_5 will cook or x_5 will not go running. For all x_5, if x_5 were a carpenter, then x_5 is running. For all x_5, if x_5 cooks, then x_5 is a carpenter. Can we infer the following from them? For all x_5, x_5 will go running if and only if x_5 will cook.
 From the fact that for all x_5, if x_5 cooks, then x_5 is a carpenter, and that for all x_5, if x_5 were a carpenter, then x_5 is running we can infer that for all x_5, if x_5 cooks, then x_5 is running. From the fact that for all x_5, x_5 will cook or x_5 will not go running we can infer that for all x_5, if x_5 goes running, then x_5 is cooking. Finally, from the fact that for all x_5, if x_5 goes running, then x_5 is cooking, and that for all x_5, if x_5 cooks, then x_5 is running we can infer that for all x_5, x_5 will go running if and only if x_5 will cook. Therefore, the answer is yes.