Consider the following premises. For all x, if x were sad, then x is climbing a mountain. There is at least one x for which x will not climb a mountain. For all x, if x climbs a mountain, then x is sad. For all x, if x were sad, then x is a musician. For all x, if x were a musician, then x is sad. Can we infer the following from them? If we can, name the inference rule being used: For all x, x is a musician.
 From the fact that for all x, if x climbs a mountain, then x is sad, and that for all x, if x were sad, then x is climbing a mountain we can infer that for all x, x will climb a mountain if and only if x is sad via universal biconditional introduction. From the fact that for all x, if x were a musician, then x is sad, and that for all x, if x were sad, then x is a musician we can infer that for all x, x is a musician if and only if x is sad via universal biconditional introduction. From the fact that for all x, x will climb a mountain if and only if x is sad, and that there is at least one x for which x will not climb a mountain we can infer that there is at least one x for which x is not sad via existential biconditional elimination. Finally, from the fact that for all x, x is a musician if and only if x is sad, and that there is at least one x for which x is not sad we can infer that there is at least one x for which x is not a musician via existential biconditional elimination, which contradicts that For all x, x is a musician. Therefore, the answer is no.