Consider the following premises. For all x_4, if x_4 plays a game, then x_4 is a lawyer. For all x_4, x_4 will listen to a song or x_4 will play a game. There is at least one x_4 for which x_4 will not listen to a song or x_4 is an office clerk. For all x_4, if x_4 were an office clerk, then x_4 is happy. For all x_4, if x_4 were happy, then x_4 is a lawyer. Can we infer the following from them? If we can, name the inference rule being used: For all x_4, x_4 is not a lawyer.
 From the fact that for all x_4, if x_4 were an office clerk, then x_4 is happy, and that for all x_4, if x_4 were happy, then x_4 is a lawyer we can infer that for all x_4, if x_4 were an office clerk, then x_4 is a lawyer via universal transitivity. From the fact that for all x_4, x_4 will listen to a song or x_4 will play a game, and that there is at least one x_4 for which x_4 will not listen to a song or x_4 is an office clerk we can infer that there is at least one x_4 for which x_4 will play a game or x_4 is an office clerk via existential resolution. Finally, from the fact that for all x_4, if x_4 plays a game, then x_4 is a lawyer, that for all x_4, if x_4 were an office clerk, then x_4 is a lawyer, and that there is at least one x_4 for which x_4 will play a game or x_4 is an office clerk we can infer that there is at least one x_4 for which x_4 is a lawyer via existential disjunction elimination, which contradicts that For all x_4, x_4 is not a lawyer. Therefore, the answer is no.