Consider the following premises. For all x_31, x_31 is a sailor if and only if x_31 is a carpenter. For all x_31, x_31 is an artist if and only if x_31 is a mechanic. For all x_31, if x_31 were a mechanic, then x_31 is an astronaut. For all x_31, if x_31 were an astronaut, then x_31 is not running. For all x_31, x_31 is a carpenter or x_31 will go running. There is at least one x_31 for which x_31 is not a sailor. Can we infer the following from them? If we can, name the inference rule being used: For all x_31, x_31 is an artist.
 From the fact that for all x_31, x_31 is a sailor if and only if x_31 is a carpenter, and that there is at least one x_31 for which x_31 is not a sailor we can infer that there is at least one x_31 for which x_31 is not a carpenter via existential biconditional elimination. From the fact that for all x_31, x_31 is a carpenter or x_31 will go running, and that there is at least one x_31 for which x_31 is not a carpenter we can infer that there is at least one x_31 for which x_31 will go running via existential disjunctive syllogism. From the fact that for all x_31, if x_31 were an astronaut, then x_31 is not running, and that there is at least one x_31 for which x_31 will go running we can infer that there is at least one x_31 for which x_31 is not an astronaut via existential modus tollens. From the fact that for all x_31, if x_31 were a mechanic, then x_31 is an astronaut, and that there is at least one x_31 for which x_31 is not an astronaut we can infer that there is at least one x_31 for which x_31 is not a mechanic via existential modus tollens. Finally, from the fact that for all x_31, x_31 is an artist if and only if x_31 is a mechanic, and that there is at least one x_31 for which x_31 is not a mechanic we can infer that there is at least one x_31 for which x_31 is not an artist via existential biconditional elimination, which contradicts that For all x_31, x_31 is an artist. Therefore, the answer is no.