Consider the following premises. For all z_4, z_4 is bored if and only if z_4 is an astronaut. For all z_4, if z_4 were sad, then z_4 is an astronaut. There is at least one z_4 for which if z_4 were bored, then z_4 is sad. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z_4 for which z_4 is bored if and only if z_4 is an astronaut.
 From the fact that there is at least one z_4 for which if z_4 were bored, then z_4 is sad, and that for all z_4, if z_4 were sad, then z_4 is an astronaut we can infer that there is at least one z_4 for which if z_4 were bored, then z_4 is an astronaut via existential transitivity. From the fact that for all z_4, z_4 is bored if and only if z_4 is an astronaut we can infer that for all z_4, if z_4 were an astronaut, then z_4 is bored via universal biconditional elimination. Finally, from the fact that there is at least one z_4 for which if z_4 were bored, then z_4 is an astronaut, and that for all z_4, if z_4 were an astronaut, then z_4 is bored we can infer that there is at least one z_4 for which z_4 is bored if and only if z_4 is an astronaut via existential biconditional introduction. Therefore, the answer is yes.