Consider the following premises. For all x_10, x_10 is a bartender if and only if x_10 is an artist. For all x_10, x_10 will not play tennis. For all x_10, if x_10 were an artist, then x_10 is not listening to a song. There is at least one x_10 for which if x_10 were a carpenter, then x_10 is a bartender. For all x_10, x_10 will play tennis or x_10 will listen to a song. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 is not a carpenter.
 From the fact that for all x_10, x_10 will play tennis or x_10 will listen to a song, and that for all x_10, x_10 will not play tennis we can infer that for all x_10, x_10 will listen to a song via universal disjunctive syllogism. From the fact that for all x_10, if x_10 were an artist, then x_10 is not listening to a song, and that for all x_10, x_10 will listen to a song we can infer that for all x_10, x_10 is not an artist via universal modus tollens. From the fact that for all x_10, x_10 is a bartender if and only if x_10 is an artist, and that for all x_10, x_10 is not an artist we can infer that for all x_10, x_10 is not a bartender via universal biconditional elimination. Finally, from the fact that there is at least one x_10 for which if x_10 were a carpenter, then x_10 is a bartender, and that for all x_10, x_10 is not a bartender we can infer that there is at least one x_10 for which x_10 is not a carpenter via existential modus tollens. Therefore, the answer is yes.