Consider the following premises. For all x_7, if x_7 reads a book, then x_7 is a bartender. For all x_7, x_7 is happy or x_7 is not a mechanic. There is at least one x_7 for which if x_7 were happy, then x_7 is reading a book. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which if x_7 were a mechanic, then x_7 is a bartender.
 From the fact that for all x_7, x_7 is happy or x_7 is not a mechanic we can infer that for all x_7, if x_7 were a mechanic, then x_7 is happy via universal rewriting. From the fact that for all x_7, if x_7 were a mechanic, then x_7 is happy, and that there is at least one x_7 for which if x_7 were happy, then x_7 is reading a book we can infer that there is at least one x_7 for which if x_7 were a mechanic, then x_7 is reading a book via existential transitivity. Finally, from the fact that there is at least one x_7 for which if x_7 were a mechanic, then x_7 is reading a book, and that for all x_7, if x_7 reads a book, then x_7 is a bartender we can infer that there is at least one x_7 for which if x_7 were a mechanic, then x_7 is a bartender via existential transitivity. Therefore, the answer is yes.