Consider the following premises. For all x_15, x_15 is sad. For all x_15, if x_15 cooks, then x_15 is a poet. For all x_15, if x_15 were sad, then x_15 is running. There is at least one x_15 for which x_15 will cook if and only if x_15 will go running. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_15 for which x_15 is a poet.
 From the fact that there is at least one x_15 for which x_15 will cook if and only if x_15 will go running we can infer that there is at least one x_15 for which if x_15 goes running, then x_15 is cooking via existential biconditional elimination. From the fact that for all x_15, if x_15 were sad, then x_15 is running, and that there is at least one x_15 for which if x_15 goes running, then x_15 is cooking we can infer that there is at least one x_15 for which if x_15 were sad, then x_15 is cooking via existential transitivity. From the fact that there is at least one x_15 for which if x_15 were sad, then x_15 is cooking, and that for all x_15, if x_15 cooks, then x_15 is a poet we can infer that there is at least one x_15 for which if x_15 were sad, then x_15 is a poet via existential transitivity. Finally, from the fact that there is at least one x_15 for which if x_15 were sad, then x_15 is a poet, and that for all x_15, x_15 is sad we can infer that there is at least one x_15 for which x_15 is a poet via existential modus ponens. Therefore, the answer is yes.