Consider the following premises. For all x_8, if x_8 were excited, then x_8 is an astronaut. For all x_8, if x_8 were an astronaut, then x_8 is excited. John is bored or Joseph is an astronaut. If John were bored, then Joseph is excited. Can we infer the following from them? If we can, name the inference rule being used: Joseph is not excited.
 From the fact that for all x_8, if x_8 were excited, then x_8 is an astronaut, and that for all x_8, if x_8 were an astronaut, then x_8 is excited we can infer that for all x_8, x_8 is excited if and only if x_8 is an astronaut via universal biconditional introduction. From the fact that for all x_8, x_8 is excited if and only if x_8 is an astronaut we can infer that Joseph is excited if and only if Joseph is an astronaut via universal instantiation. From the fact that Joseph is excited if and only if Joseph is an astronaut we can infer that if Joseph were an astronaut, then Joseph is excited via biconditional elimination. Finally, from the fact that if John were bored, then Joseph is excited, that if Joseph were an astronaut, then Joseph is excited, and that John is bored or Joseph is an astronaut we can infer that Joseph is excited via disjunction elimination, which contradicts that Joseph is not excited. Therefore, the answer is no.