Consider the following premises. For all y, if y were curious, then y is sleeping. For all y, if y cooks, then y is driving a car. There is at least one y for which if y sleeps, then y is cooking. For all y, if y drives a car, then y is curious. Can we infer the following from them? If we can, name the inference rule being used: For all y, if is not true that y will drive a car if and only if y is curious.
 From the fact that there is at least one y for which if y sleeps, then y is cooking, and that for all y, if y cooks, then y is driving a car we can infer that there is at least one y for which if y sleeps, then y is driving a car via existential transitivity. From the fact that for all y, if y were curious, then y is sleeping, and that there is at least one y for which if y sleeps, then y is driving a car we can infer that there is at least one y for which if y were curious, then y is driving a car via existential transitivity. Finally, from the fact that for all y, if y drives a car, then y is curious, and that there is at least one y for which if y were curious, then y is driving a car we can infer that there is at least one y for which y will drive a car if and only if y is curious via existential biconditional introduction, which contradicts that For all y, if is not true that y will drive a car if and only if y is curious. Therefore, the answer is no.