Consider the following premises. For all z, z is not sad. For all z, if z makes tea, then z is sad. For all z, z will make tea if and only if z is a bartender. There is at least one z for which if z were sad, then z is a bartender. 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 will not make tea.
 From the fact that for all z, z will make tea if and only if z is a bartender we can infer that for all z, if z were a bartender, then z is making tea via universal biconditional elimination. From the fact that there is at least one z for which if z were sad, then z is a bartender, and that for all z, if z were a bartender, then z is making tea we can infer that there is at least one z for which if z were sad, then z is making tea via existential transitivity. From the fact that for all z, if z makes tea, then z is sad, and that there is at least one z for which if z were sad, then z is making tea we can infer that there is at least one z for which z will make tea if and only if z is sad via existential biconditional introduction. Finally, from the fact that there is at least one z for which z will make tea if and only if z is sad, and that for all z, z is not sad we can infer that there is at least one z for which z will not make tea via existential biconditional elimination. Therefore, the answer is yes.