Consider the following premises. For all x_12, if x_12 were a politician, then x_12 is an astronaut. For all x_12, x_12 is a server or x_12 is a politician. For all x_12, if x_12 were a server, then x_12 is bored. For all x_12, x_12 is an astronaut if and only if x_12 is bored. There is at least one x_12 for which if x_12 were excited, then x_12 is not an astronaut. Can we infer the following from them? If we can, name the inference rule being used: For all x_12, x_12 is excited.
 From the fact that for all x_12, x_12 is an astronaut if and only if x_12 is bored we can infer that for all x_12, if x_12 were bored, then x_12 is an astronaut via universal biconditional elimination. From the fact that for all x_12, if x_12 were a server, then x_12 is bored, and that for all x_12, if x_12 were bored, then x_12 is an astronaut we can infer that for all x_12, if x_12 were a server, then x_12 is an astronaut via universal transitivity. From the fact that for all x_12, if x_12 were a server, then x_12 is an astronaut, that for all x_12, if x_12 were a politician, then x_12 is an astronaut, and that for all x_12, x_12 is a server or x_12 is a politician we can infer that for all x_12, x_12 is an astronaut via universal disjunction elimination. Finally, from the fact that there is at least one x_12 for which if x_12 were excited, then x_12 is not an astronaut, and that for all x_12, x_12 is an astronaut we can infer that there is at least one x_12 for which x_12 is not excited via existential modus tollens, which contradicts that For all x_12, x_12 is excited. Therefore, the answer is no.