Consider the following premises. For all z, z is an astronaut if and only if z will go running. For all z, if z goes running, then z is an astronaut. Can we infer the following from them? For all z, z will go running if and only if z is an astronaut.
 From the fact that for all z, z is an astronaut if and only if z will go running we can infer that for all z, if z were an astronaut, then z is running. Finally, from the fact that for all z, if z goes running, then z is an astronaut, and that for all z, if z were an astronaut, then z is running we can infer that for all z, z will go running if and only if z is an astronaut. Therefore, the answer is yes.