Consider the following premises. If William climbs a mountain, then it is cloudy. It is not cloudy. If John were an artist, then William is climbing a mountain. For all x_7, x_7 is a writer if and only if x_7 is an artist. Can we infer the following from them? John is not a writer.
 From the fact that if William climbs a mountain, then it is cloudy, and that it is not cloudy we can infer that William is not climbing a mountain. From the fact that for all x_7, x_7 is a writer if and only if x_7 is an artist we can infer that John is a writer if and only if John is an artist. From the fact that if John were an artist, then William is climbing a mountain, and that William is not climbing a mountain we can infer that John is not an artist. Finally, from the fact that John is a writer if and only if John is an artist, and that John is not an artist we can infer that John is not a writer. Therefore, the answer is yes.