Consider the following premises. Patricia is a musician. For all x_5, x_5 is a musician if and only if x_5 will climb a mountain. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which x_5 will climb a mountain.
 From the fact that Patricia is a musician we can infer that there is at least one x_5 for which x_5 is a musician via existential generalization. From the fact that for all x_5, x_5 is a musician if and only if x_5 will climb a mountain we can infer that for all x_5, if x_5 were a musician, then x_5 is climbing a mountain via universal biconditional elimination. Finally, from the fact that for all x_5, if x_5 were a musician, then x_5 is climbing a mountain, and that there is at least one x_5 for which x_5 is a musician we can infer that there is at least one x_5 for which x_5 will climb a mountain via existential modus ponens. Therefore, the answer is yes.