Consider the following premises. There is at least one x_7 for which x_7 is a scientist or x_7 is not happy. For all x_7, if x_7 were a scientist, then x_7 is a carpenter. For all x_7, if x_7 were a carpenter, then x_7 is happy. Can we infer the following from them? If we can, name the inference rule being used: For all x_7, if is not true that x_7 is happy if and only if x_7 is a scientist.
 From the fact that there is at least one x_7 for which x_7 is a scientist or x_7 is not happy we can infer that there is at least one x_7 for which if x_7 were happy, then x_7 is a scientist via existential rewriting. From the fact that for all x_7, if x_7 were a scientist, then x_7 is a carpenter, and that for all x_7, if x_7 were a carpenter, then x_7 is happy we can infer that for all x_7, if x_7 were a scientist, then x_7 is happy via universal transitivity. Finally, from the fact that there is at least one x_7 for which if x_7 were happy, then x_7 is a scientist, and that for all x_7, if x_7 were a scientist, then x_7 is happy we can infer that there is at least one x_7 for which x_7 is happy if and only if x_7 is a scientist via existential biconditional introduction, which contradicts that For all x_7, if is not true that x_7 is happy if and only if x_7 is a scientist. Therefore, the answer is no.