Consider the following premises. For all z, if z listens to a song, then z is excited. For all z, z is a janitor or z will play squash. There is at least one z for which if z plays squash, then z is listening to a song. For all z, z is excited or z is not a carpenter. For all z, z is not a janitor or z is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which z is excited.
 From the fact that for all z, z is a janitor or z will play squash, and that for all z, z is not a janitor or z is a carpenter we can infer that for all z, z will play squash or z is a carpenter via universal resolution. From the fact that for all z, z is excited or z is not a carpenter we can infer that for all z, if z were a carpenter, then z is excited via universal rewriting. From the fact that there is at least one z for which if z plays squash, then z is listening to a song, and that for all z, if z listens to a song, then z is excited we can infer that there is at least one z for which if z plays squash, then z is excited via existential transitivity. Finally, from the fact that there is at least one z for which if z plays squash, then z is excited, that for all z, if z were a carpenter, then z is excited, and that for all z, z will play squash or z is a carpenter we can infer that there is at least one z for which z is excited via existential disjunction elimination. Therefore, the answer is yes.