Consider the following premises. For all y, if y were excited, then y is happy. For all y, y is excited or y is not happy. For all y, if y were happy, then y is excited. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y for which if is not true that y is happy if and only if y is excited.
 From the fact that for all y, if y were excited, then y is happy, and that for all y, if y were happy, then y is excited we can infer that for all y, y is excited if and only if y is happy via universal biconditional introduction. From the fact that for all y, y is excited if and only if y is happy we can infer that for all y, if y were excited, then y is happy via universal biconditional elimination. From the fact that for all y, y is excited or y is not happy we can infer that for all y, if y were happy, then y is excited via universal rewriting. Finally, from the fact that for all y, if y were happy, then y is excited, and that for all y, if y were excited, then y is happy we can infer that for all y, y is happy if and only if y is excited via universal biconditional introduction, which contradicts that There is at least one y for which if is not true that y is happy if and only if y is excited. Therefore, the answer is no.