Consider the following premises. For all x_1, x_1 is an artist. For all x_1, if x_1 were an artist, then x_1 is playing a game. There is at least one x_1 for which x_1 is poor. Can we infer the following from them? If we can, name the inference rule being used: For all x_1, either x_1 is poor or x_1 will play a game are not true.
 From the fact that for all x_1, if x_1 were an artist, then x_1 is playing a game, and that for all x_1, x_1 is an artist we can infer that for all x_1, x_1 will play a game via universal modus ponens. Finally, from the fact that there is at least one x_1 for which x_1 is poor, and that for all x_1, x_1 will play a game we can infer that there is at least one x_1 for which x_1 is poor and x_1 will play a game via existential conjunction, which contradicts that For all x_1, either x_1 is poor or x_1 will play a game are not true. Therefore, the answer is no.