Consider the following premises. For all x_10, if x_10 climbs a mountain, then x_10 is an artist. For all x_10, x_10 will read a book. There is at least one x_10 for which x_10 will climb a mountain or x_10 is a musician. For all x_10, if x_10 were a musician, then x_10 is writing a letter. For all x_10, if x_10 were an artist, then x_10 is writing a letter. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, either x_10 will read a book or x_10 will write a letter are not true.
 From the fact that for all x_10, if x_10 climbs a mountain, then x_10 is an artist, and that for all x_10, if x_10 were an artist, then x_10 is writing a letter we can infer that for all x_10, if x_10 climbs a mountain, then x_10 is writing a letter via universal transitivity. From the fact that for all x_10, if x_10 climbs a mountain, then x_10 is writing a letter, that for all x_10, if x_10 were a musician, then x_10 is writing a letter, and that there is at least one x_10 for which x_10 will climb a mountain or x_10 is a musician we can infer that there is at least one x_10 for which x_10 will write a letter via existential disjunction elimination. Finally, from the fact that for all x_10, x_10 will read a book, and that there is at least one x_10 for which x_10 will write a letter we can infer that there is at least one x_10 for which x_10 will read a book and x_10 will write a letter via existential conjunction, which contradicts that For all x_10, either x_10 will read a book or x_10 will write a letter are not true. Therefore, the answer is no.