Consider the following premises. For all x_8, x_8 will write a letter if and only if x_8 is a carpenter. For all x_8, x_8 is not a carpenter. For all x_8, x_8 will write a letter if and only if x_8 is an astronaut. Can we infer the following from them? If we can, name the inference rule being used: For all x_8, x_8 is not an astronaut.
 From the fact that for all x_8, x_8 will write a letter if and only if x_8 is a carpenter, and that for all x_8, x_8 is not a carpenter we can infer that for all x_8, x_8 will not write a letter via universal biconditional elimination. From the fact that for all x_8, x_8 will write a letter if and only if x_8 is an astronaut we can infer that for all x_8, if x_8 were an astronaut, then x_8 is writing a letter via universal biconditional elimination. Finally, from the fact that for all x_8, if x_8 were an astronaut, then x_8 is writing a letter, and that for all x_8, x_8 will not write a letter we can infer that for all x_8, x_8 is not an astronaut via universal modus tollens. Therefore, the answer is yes.