Consider the following premises. For all x_13, x_13 is a mechanic if and only if x_13 will climb a mountain. For all x_13, x_13 is a bartender. There is at least one x_13 for which x_13 will climb a mountain. 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 mechanic or x_13 is a bartender are not true.
 From the fact that for all x_13, x_13 is a mechanic if and only if x_13 will climb a mountain we can infer that for all x_13, if x_13 climbs a mountain, then x_13 is a mechanic via universal biconditional elimination. From the fact that for all x_13, if x_13 climbs a mountain, then x_13 is a mechanic, and that there is at least one x_13 for which x_13 will climb a mountain we can infer that there is at least one x_13 for which x_13 is a mechanic via existential modus ponens. Finally, from the fact that there is at least one x_13 for which x_13 is a mechanic, and that for all x_13, x_13 is a bartender we can infer that some mechanics are bartenders via existential conjunction, which contradicts that For all x_13, either x_13 is a mechanic or x_13 is a bartender are not true. Therefore, the answer is no.