Consider the following premises. For all x_12, x_12 is a carpenter or x_12 will play a game. For all x_12, x_12 is a mechanic and x_12 is a scientist. There is at least one x_12 for which x_12 is not a carpenter. 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 will play a game and x_12 is a mechanic.
 From the fact that for all x_12, x_12 is a mechanic and x_12 is a scientist we can infer that for all x_12, x_12 is a mechanic via universal simplification. From the fact that for all x_12, x_12 is a carpenter or x_12 will play a game, 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 will play a game via existential disjunctive syllogism. Finally, from the fact that there is at least one x_12 for which x_12 will play a game, and that for all x_12, x_12 is a mechanic we can infer that there is at least one x_12 for which x_12 will play a game and x_12 is a mechanic via existential conjunction. Therefore, the answer is yes.