Consider the following premises. For all z_3, if z_3 were a cashier, then z_3 is making tea. For all z_3, if z_3 were a cashier, then z_3 is listening to a song. For all z_3, z_3 will listen to a song or z_3 will not work. There is at least one z_3 for which if z_3 listens to a song, then z_3 is a cashier. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z_3 for which if z_3 works, then z_3 is making tea.
 From the fact that there is at least one z_3 for which if z_3 listens to a song, then z_3 is a cashier, and that for all z_3, if z_3 were a cashier, then z_3 is listening to a song we can infer that there is at least one z_3 for which z_3 will listen to a song if and only if z_3 is a cashier via existential biconditional introduction. From the fact that for all z_3, z_3 will listen to a song or z_3 will not work we can infer that for all z_3, if z_3 works, then z_3 is listening to a song via universal rewriting. From the fact that there is at least one z_3 for which z_3 will listen to a song if and only if z_3 is a cashier we can infer that there is at least one z_3 for which if z_3 listens to a song, then z_3 is a cashier via existential biconditional elimination. From the fact that there is at least one z_3 for which if z_3 listens to a song, then z_3 is a cashier, and that for all z_3, if z_3 were a cashier, then z_3 is making tea we can infer that there is at least one z_3 for which if z_3 listens to a song, then z_3 is making tea via existential transitivity. Finally, from the fact that for all z_3, if z_3 works, then z_3 is listening to a song, and that there is at least one z_3 for which if z_3 listens to a song, then z_3 is making tea we can infer that there is at least one z_3 for which if z_3 works, then z_3 is making tea via existential transitivity. Therefore, the answer is yes.