Consider the following premises. For all x_14, x_14 will climb a mountain if and only if x_14 will write a letter. There is at least one x_14 for which if x_14 were a writer, then x_14 is writing a letter. For all x_14, if x_14 writes a letter, then x_14 is a writer. For all x_14, if x_14 were a writer, then x_14 is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: For all x_14, if is not true that x_14 will write a letter if and only if x_14 is a writer.
That contradicts the premises. Therefore the answer is no.