Consider the following premises. Joseph is playing tennis if and only if Joseph is climbing a mountain. If Joseph plays tennis, then Joseph is climbing a mountain. If Elisabeth were a cashier, then Joseph is playing tennis. Joseph is climbing a mountain or Elisabeth is a cashier. Can we infer the following from them? Joseph is not playing tennis.
 From the fact that Joseph is playing tennis if and only if Joseph is climbing a mountain we can infer that if Joseph climbs a mountain, then Joseph is playing tennis. From the fact that if Joseph plays tennis, then Joseph is climbing a mountain, and that if Joseph climbs a mountain, then Joseph is playing tennis we can infer that Joseph is playing tennis if and only if Joseph is climbing a mountain. From the fact that Joseph is playing tennis if and only if Joseph is climbing a mountain we can infer that if Joseph climbs a mountain, then Joseph is playing tennis. Finally, from the fact that if Joseph climbs a mountain, then Joseph is playing tennis, that if Elisabeth were a cashier, then Joseph is playing tennis, and that Joseph is climbing a mountain or Elisabeth is a cashier we can infer that Joseph is playing tennis, which contradicts that Joseph is not playing tennis. Therefore, the answer is no.