Consider the following premises. For all z_1, if z_1 takes a plane, then z_1 is writing a letter. For all z_1, z_1 is an astronaut or z_1 will not write a letter. For all z_1, if z_1 were an astronaut, then z_1 is taking a plane. Can we infer the following from them? If we can, name the inference rule being used: For all z_1, z_1 will write a letter if and only if z_1 is an astronaut.
 From the fact that for all z_1, if z_1 were an astronaut, then z_1 is taking a plane, and that for all z_1, if z_1 takes a plane, then z_1 is writing a letter we can infer that for all z_1, if z_1 were an astronaut, then z_1 is writing a letter via universal transitivity. From the fact that for all z_1, z_1 is an astronaut or z_1 will not write a letter we can infer that for all z_1, if z_1 writes a letter, then z_1 is an astronaut via universal rewriting. Finally, from the fact that for all z_1, if z_1 writes a letter, then z_1 is an astronaut, and that for all z_1, if z_1 were an astronaut, then z_1 is writing a letter we can infer that for all z_1, z_1 will write a letter if and only if z_1 is an astronaut via universal biconditional introduction. Therefore, the answer is yes.