Consider the following premises. For all x_6, if x_6 were a lawyer, then x_6 is cooking. For all x_6, if x_6 takes a plane, then x_6 is a lawyer. For all x_6, if x_6 listens to a song, then x_6 is a carpenter. There is at least one x_6 for which if x_6 were a carpenter, then x_6 is taking a plane. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which if x_6 listens to a song, then x_6 is cooking.
 From the fact that there is at least one x_6 for which if x_6 were a carpenter, then x_6 is taking a plane, and that for all x_6, if x_6 takes a plane, then x_6 is a lawyer we can infer that there is at least one x_6 for which if x_6 were a carpenter, then x_6 is a lawyer via existential transitivity. From the fact that for all x_6, if x_6 listens to a song, then x_6 is a carpenter, and that there is at least one x_6 for which if x_6 were a carpenter, then x_6 is a lawyer we can infer that there is at least one x_6 for which if x_6 listens to a song, then x_6 is a lawyer via existential transitivity. Finally, from the fact that there is at least one x_6 for which if x_6 listens to a song, then x_6 is a lawyer, and that for all x_6, if x_6 were a lawyer, then x_6 is cooking we can infer that there is at least one x_6 for which if x_6 listens to a song, then x_6 is cooking via existential transitivity. Therefore, the answer is yes.