Consider the following premises. For all x, if x were a carpenter, then x is poor. For all x, if x were poor, then x is making tea. There is at least one x for which if x makes tea, then x is a carpenter. For all x, x is not poor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x will not make tea.
 From the fact that there is at least one x for which if x makes tea, then x is a carpenter, and that for all x, if x were a carpenter, then x is poor we can infer that there is at least one x for which if x makes tea, then x is poor via existential transitivity. From the fact that for all x, if x were poor, then x is making tea, and that there is at least one x for which if x makes tea, then x is poor we can infer that there is at least one x for which x is poor if and only if x will make tea via existential biconditional introduction. Finally, from the fact that there is at least one x for which x is poor if and only if x will make tea, and that for all x, x is not poor we can infer that there is at least one x for which x will not make tea via existential biconditional elimination. Therefore, the answer is yes.