Consider the following premises. For all x_4, x_4 will climb a mountain or x_4 will not write a letter. For all x_4, x_4 is not a scientist. For all x_4, x_4 will climb a mountain if and only if x_4 is a scientist. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_4 for which x_4 will write a letter.
 From the fact that for all x_4, x_4 will climb a mountain if and only if x_4 is a scientist, and that for all x_4, x_4 is not a scientist we can infer that for all x_4, x_4 will not climb a mountain via universal biconditional elimination. From the fact that for all x_4, x_4 will climb a mountain or x_4 will not write a letter we can infer that for all x_4, if x_4 writes a letter, then x_4 is climbing a mountain via universal rewriting. Finally, from the fact that for all x_4, if x_4 writes a letter, then x_4 is climbing a mountain, and that for all x_4, x_4 will not climb a mountain we can infer that for all x_4, x_4 will not write a letter via universal modus tollens, which contradicts that There is at least one x_4 for which x_4 will write a letter. Therefore, the answer is no.