Consider the following premises. For all x_7, x_7 will play a game or x_7 is rich. There is at least one x_7 for which x_7 will not play a game or x_7 will drive a car. For all x_7, x_7 is rich if and only if x_7 is a mechanic. For all x_7, if x_7 drives a car, then x_7 is a mechanic. Can we infer the following from them? If we can, name the inference rule being used: For all x_7, x_7 is not a mechanic.
 From the fact that for all x_7, x_7 will play a game or x_7 is rich, and that there is at least one x_7 for which x_7 will not play a game or x_7 will drive a car we can infer that there is at least one x_7 for which x_7 is rich or x_7 will drive a car via existential resolution. From the fact that for all x_7, x_7 is rich if and only if x_7 is a mechanic we can infer that for all x_7, if x_7 were rich, then x_7 is a mechanic via universal biconditional elimination. Finally, from the fact that for all x_7, if x_7 were rich, then x_7 is a mechanic, that for all x_7, if x_7 drives a car, then x_7 is a mechanic, and that there is at least one x_7 for which x_7 is rich or x_7 will drive a car we can infer that there is at least one x_7 for which x_7 is a mechanic via existential disjunction elimination, which contradicts that For all x_7, x_7 is not a mechanic. Therefore, the answer is no.