Consider the following premises. For all x_13, x_13 will write a letter. For all x_13, if x_13 were a musician, then x_13 is working. There is at least one x_13 for which if x_13 writes a letter, then x_13 is a poet. For all x_13, if x_13 were bored, then x_13 is working. For all x_13, x_13 is a musician or x_13 is bored. Can we infer the following from them? If we can, name the inference rule being used: For all x_13, either x_13 is a poet or x_13 will work are not true.
 From the fact that for all x_13, if x_13 were a musician, then x_13 is working, that for all x_13, if x_13 were bored, then x_13 is working, and that for all x_13, x_13 is a musician or x_13 is bored we can infer that for all x_13, x_13 will work via universal disjunction elimination. From the fact that there is at least one x_13 for which if x_13 writes a letter, then x_13 is a poet, and that for all x_13, x_13 will write a letter we can infer that there is at least one x_13 for which x_13 is a poet via existential modus ponens. Finally, from the fact that there is at least one x_13 for which x_13 is a poet, and that for all x_13, x_13 will work we can infer that some poets will work via existential conjunction, which contradicts that For all x_13, either x_13 is a poet or x_13 will work are not true. Therefore, the answer is no.