Consider the following premises. For all y, y is a poet or y will not cook. For all y, if y were a mechanic, then y is cooking. For all y, if y cooks, then y is a mechanic. There is at least one y for which if y were a poet, then y is a mechanic. Can we infer the following from them? There is at least one y for which y is a mechanic if and only if y is a poet.
 From the fact that for all y, if y cooks, then y is a mechanic, and that for all y, if y were a mechanic, then y is cooking we can infer that for all y, y will cook if and only if y is a mechanic. From the fact that for all y, y will cook if and only if y is a mechanic we can infer that for all y, if y were a mechanic, then y is cooking. From the fact that for all y, y is a poet or y will not cook we can infer that for all y, if y cooks, then y is a poet. From the fact that for all y, if y were a mechanic, then y is cooking, and that for all y, if y cooks, then y is a poet we can infer that for all y, if y were a mechanic, then y is a poet. Finally, from the fact that for all y, if y were a mechanic, then y is a poet, and that there is at least one y for which if y were a poet, then y is a mechanic we can infer that there is at least one y for which y is a mechanic if and only if y is a poet. Therefore, the answer is yes.