Consider the following premises. There is at least one x for which if x were a politician, then x is cooking. For all x, x is a politician or x will write a letter. For all x, if x writes a letter, then x is cooking. For all x, if x cooks, then x is a politician. 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 cook.
 From the fact that there is at least one x for which if x were a politician, then x is cooking, and that for all x, if x cooks, then x is a politician we can infer that there is at least one x for which x is a politician if and only if x will cook via existential biconditional introduction. From the fact that there is at least one x for which x is a politician if and only if x will cook we can infer that there is at least one x for which if x were a politician, then x is cooking via existential biconditional elimination. Finally, from the fact that there is at least one x for which if x were a politician, then x is cooking, that for all x, if x writes a letter, then x is cooking, and that for all x, x is a politician or x will write a letter we can infer that there is at least one x for which x will cook via existential disjunction elimination. Therefore, the answer is yes.