Consider the following premises. For all y_2, y_2 will sleep. For all y_2, y_2 is a mechanic. For all y_2, y_2 is a mechanic if and only if y_2 is a lawyer. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y_2 for which y_2 is not a lawyer.
 From the fact that for all y_2, y_2 is a mechanic if and only if y_2 is a lawyer we can infer that for all y_2, if y_2 were a mechanic, then y_2 is a lawyer via universal biconditional elimination. From the fact that for all y_2, y_2 is a mechanic, and that for all y_2, y_2 will sleep we can infer that for all y_2, y_2 is a mechanic and y_2 will sleep via universal conjunction. From the fact that for all y_2, y_2 is a mechanic and y_2 will sleep we can infer that for all y_2, y_2 is a mechanic via universal simplification. Finally, from the fact that for all y_2, if y_2 were a mechanic, then y_2 is a lawyer, and that for all y_2, y_2 is a mechanic we can infer that for all y_2, y_2 is a lawyer via universal modus ponens, which contradicts that There is at least one y_2 for which y_2 is not a lawyer. Therefore, the answer is no.