Consider the following premises. For all y, if y goes running, then y is climbing a mountain. For all y, if y goes running, then y is reading a book. For all y, if y reads a book, then y is running. There is at least one y for which y will not climb a mountain. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y for which y will not read a book.
 From the fact that for all y, if y reads a book, then y is running, and that for all y, if y goes running, then y is reading a book we can infer that for all y, y will read a book if and only if y will go running via universal biconditional introduction. From the fact that for all y, y will read a book if and only if y will go running we can infer that for all y, if y reads a book, then y is running via universal biconditional elimination. From the fact that for all y, if y reads a book, then y is running, and that for all y, if y goes running, then y is climbing a mountain we can infer that for all y, if y reads a book, then y is climbing a mountain via universal transitivity. Finally, from the fact that for all y, if y reads a book, then y is climbing a mountain, and that there is at least one y for which y will not climb a mountain we can infer that there is at least one y for which y will not read a book via existential modus tollens. Therefore, the answer is yes.