Consider the following premises. There is at least one x_12 for which if x_12 were a carpenter, then x_12 is not a musician. For all x_12, x_12 is a carpenter or x_12 is rich. For all x_12, x_12 is a musician. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_12 for which x_12 is rich or x_12 is fast.
 From the fact that there is at least one x_12 for which if x_12 were a carpenter, then x_12 is not a musician, and that for all x_12, x_12 is a musician we can infer that there is at least one x_12 for which x_12 is not a carpenter via existential modus tollens. From the fact that for all x_12, x_12 is a carpenter or x_12 is rich, and that there is at least one x_12 for which x_12 is not a carpenter we can infer that there is at least one x_12 for which x_12 is rich via existential disjunctive syllogism. Finally, from the fact that there is at least one x_12 for which x_12 is rich we can infer that there is at least one x_12 for which x_12 is rich or x_12 is fast via existential addition. Therefore, the answer is yes.