Consider the following premises. For all y, if y were a carpenter, then y is a mechanic. For all y, y will not write a letter. For all y, if y writes a letter, then y is a mechanic. There is at least one y for which if y were a mechanic, then y is writing a letter. Can we infer the following from them? If we can, name the inference rule being used: For all y, y is a carpenter.
 From the fact that there is at least one y for which if y were a mechanic, then y is writing a letter, and that for all y, if y writes a letter, then y is a mechanic we can infer that there is at least one y for which y is a mechanic if and only if y will write a letter via existential biconditional introduction. From the fact that there is at least one y for which y is a mechanic if and only if y will write a letter, and that for all y, y will not write a letter we can infer that there is at least one y for which y is not a mechanic via existential biconditional elimination. Finally, from the fact that for all y, if y were a carpenter, then y is a mechanic, and that there is at least one y for which y is not a mechanic we can infer that there is at least one y for which y is not a carpenter via existential modus tollens, which contradicts that For all y, y is a carpenter. Therefore, the answer is no.