Consider the following premises. For all y, if y were a carpenter, then y is sad. For all y, if y were sad, then y is playing tennis. There is at least one y for which if y were happy, then y is a carpenter. For all y, if y drives a car, then y is happy. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y for which if y drives a car, then y is playing tennis.
 From the fact that there is at least one y for which if y were happy, then y is a carpenter, and that for all y, if y were a carpenter, then y is sad we can infer that there is at least one y for which if y were happy, then y is sad via existential transitivity. From the fact that for all y, if y drives a car, then y is happy, and that there is at least one y for which if y were happy, then y is sad we can infer that there is at least one y for which if y drives a car, then y is sad via existential transitivity. Finally, from the fact that there is at least one y for which if y drives a car, then y is sad, and that for all y, if y were sad, then y is playing tennis we can infer that there is at least one y for which if y drives a car, then y is playing tennis via existential transitivity. Therefore, the answer is yes.