Consider the following premises. William is not a lawyer. For all x_8, if x_8 were an astronaut, then x_8 is a lawyer. William is an astronaut or it is cloudy. For all x_8, if x_8 were a lawyer, then x_8 is an astronaut. Can we infer the following from them? If we can, name the inference rule being used: It is not cloudy.
 From the fact that for all x_8, if x_8 were a lawyer, then x_8 is an astronaut, and that for all x_8, if x_8 were an astronaut, then x_8 is a lawyer we can infer that for all x_8, x_8 is a lawyer if and only if x_8 is an astronaut via universal biconditional introduction. From the fact that for all x_8, x_8 is a lawyer if and only if x_8 is an astronaut we can infer that William is a lawyer if and only if William is an astronaut via universal instantiation. From the fact that William is a lawyer if and only if William is an astronaut, and that William is not a lawyer we can infer that William is not an astronaut via biconditional elimination. Finally, from the fact that William is an astronaut or it is cloudy, and that William is not an astronaut we can infer that it is cloudy via disjunctive syllogism, which contradicts that It is not cloudy. Therefore, the answer is no.