Consider the following premises. For all x_15, if x_15 were a scientist, then x_15 is a musician. William is a scientist or Mary is not writing a letter. If William were a musician, then Mary is writing a letter. Can we infer the following from them? Mary is writing a letter if and only if William is a scientist.
 From the fact that for all x_15, if x_15 were a scientist, then x_15 is a musician we can infer that if William were a scientist, then William is a musician. From the fact that if William were a scientist, then William is a musician, and that if William were a musician, then Mary is writing a letter we can infer that if William were a scientist, then Mary is writing a letter. From the fact that William is a scientist or Mary is not writing a letter we can infer that if Mary writes a letter, then William is a scientist. Finally, from the fact that if Mary writes a letter, then William is a scientist, and that if William were a scientist, then Mary is writing a letter we can infer that Mary is writing a letter if and only if William is a scientist. Therefore, the answer is yes.