Consider the following premises. For all x_10, if x_10 were a lawyer, then x_10 is climbing a mountain. For all x_10, if x_10 were a bartender, then x_10 is climbing a mountain. There is at least one x_10 for which x_10 is a bartender. For all x_10, if x_10 climbs a mountain, then x_10 is an artist. 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 there is at least one x_10 for which x_10 is a bartender we can infer that there is at least one x_10 for which x_10 is a bartender or x_10 is a lawyer via existential addition. From the fact that for all x_10, if x_10 were a bartender, then x_10 is climbing a mountain, that for all x_10, if x_10 were a lawyer, then x_10 is climbing a mountain, and that there is at least one x_10 for which x_10 is a bartender or x_10 is a lawyer we can infer that there is at least one x_10 for which x_10 will climb a mountain via existential disjunction elimination. Finally, from the fact that for all x_10, if x_10 climbs a mountain, then x_10 is an artist, and that there is at least one x_10 for which x_10 will climb a mountain we can infer that there is at least one x_10 for which x_10 is an artist via existential modus ponens, which contradicts that For all x_10, x_10 is not an artist. Therefore, the answer is no.