Consider the following premises. For all z, if z listens to a song, then z is a janitor. For all z, if z were an electrician, then z is playing a game. For all z, if z plays squash, then z is listening to a song. There is at least one z for which if z were a janitor, then z is an electrician. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which if z plays squash, then z is playing a game.
 From the fact that for all z, 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 a janitor we can infer that for all z, if z plays squash, then z is a janitor via universal transitivity. From the fact that for all z, if z plays squash, then z is a janitor, and that there is at least one z for which if z were a janitor, then z is an electrician we can infer that there is at least one z for which if z plays squash, then z is an electrician via existential transitivity. Finally, from the fact that there is at least one z for which if z plays squash, then z is an electrician, and that for all z, if z were an electrician, then z is playing a game we can infer that there is at least one z for which if z plays squash, then z is playing a game via existential transitivity. Therefore, the answer is yes.