Consider the following premises. For all x_3, if x_3 were an artist, then x_3 is working. For all x_3, if x_3 works, then x_3 is playing a game. For all x_3, x_3 is not an artist. There is at least one x_3 for which x_3 is an astronaut if and only if x_3 will work. For all x_3, if x_3 plays a game, then x_3 is an artist. Can we infer the following from them? If we can, name the inference rule being used: For all x_3, x_3 is an astronaut.
 From the fact that for all x_3, if x_3 works, then x_3 is playing a game, and that for all x_3, if x_3 plays a game, then x_3 is an artist we can infer that for all x_3, if x_3 works, then x_3 is an artist via universal transitivity. From the fact that for all x_3, if x_3 works, then x_3 is an artist, and that for all x_3, if x_3 were an artist, then x_3 is working we can infer that for all x_3, x_3 will work if and only if x_3 is an artist via universal biconditional introduction. From the fact that for all x_3, x_3 will work if and only if x_3 is an artist, and that for all x_3, x_3 is not an artist we can infer that for all x_3, x_3 will not work via universal biconditional elimination. Finally, from the fact that there is at least one x_3 for which x_3 is an astronaut if and only if x_3 will work, and that for all x_3, x_3 will not work we can infer that there is at least one x_3 for which x_3 is not an astronaut via existential biconditional elimination, which contradicts that For all x_3, x_3 is an astronaut. Therefore, the answer is no.