Consider the following premises. For all x_10, if x_10 were rich, then x_10 is writing a letter. For all x_10, x_10 is a poet or x_10 is an artist. There is at least one x_10 for which x_10 will not write a letter. For all x_10, x_10 is a poet if and only if x_10 is rich. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, x_10 is not an artist.
 From the fact that for all x_10, if x_10 were rich, then x_10 is writing a letter, and that there is at least one x_10 for which x_10 will not write a letter we can infer that there is at least one x_10 for which x_10 is not rich via existential modus tollens. From the fact that for all x_10, x_10 is a poet if and only if x_10 is rich, and that there is at least one x_10 for which x_10 is not rich we can infer that there is at least one x_10 for which x_10 is not a poet via existential biconditional elimination. Finally, from the fact that for all x_10, x_10 is a poet or x_10 is an artist, and that there is at least one x_10 for which x_10 is not a poet we can infer that there is at least one x_10 for which x_10 is an artist via existential disjunctive syllogism, which contradicts that For all x_10, x_10 is not an artist. Therefore, the answer is no.