Consider the following premises. For all x, if x were a mechanic, then x is rich. For all x, if x were a mechanic, then x is sleeping. There is at least one x for which if x sleeps, then x is a mechanic. For all x, x is not rich. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x will not sleep.
 From the fact that there is at least one x for which if x sleeps, then x is a mechanic, and that for all x, if x were a mechanic, then x is sleeping we can infer that there is at least one x for which x will sleep if and only if x is a mechanic via existential biconditional introduction. From the fact that for all x, if x were a mechanic, then x is rich, and that for all x, x is not rich we can infer that for all x, x is not a mechanic via universal modus tollens. Finally, from the fact that there is at least one x for which x will sleep if and only if x is a mechanic, and that for all x, x is not a mechanic we can infer that there is at least one x for which x will not sleep via existential biconditional elimination. Therefore, the answer is yes.