Consider the following premises. For all x_10, x_10 is a writer. For all x_10, x_10 will make tea. For all x_10, x_10 is an artist. There is at least one x_10 for which if x_10 were an artist, then x_10 is not a writer. 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 will write a letter.
 From the fact that there is at least one x_10 for which if x_10 were an artist, then x_10 is not a writer, and that for all x_10, x_10 is a writer we can infer that there is at least one x_10 for which x_10 is not an artist via existential modus tollens. From the fact that for all x_10, x_10 is an artist, and that for all x_10, x_10 will make tea we can infer that for all x_10, x_10 is an artist and x_10 will make tea via universal conjunction. From the fact that for all x_10, x_10 is an artist and x_10 will make tea we can infer that for all x_10, x_10 is an artist via universal simplification. From the fact that for all x_10, x_10 is an artist we can infer that for all x_10, x_10 is an artist or x_10 will write a letter via universal addition. Finally, from the fact that for all x_10, x_10 is an artist or x_10 will write a letter, and that there is at least one x_10 for which x_10 is not an artist we can infer that there is at least one x_10 for which x_10 will write a letter via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.