Consider the following premises. For all x_4, if x_4 were a doctor, then x_4 is an astronaut. For all x_4, x_4 is a doctor if and only if x_4 will play squash. For all x_4, if x_4 were an astronaut, then x_4 is playing squash. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_4 for which the fact that x_4 were a doctor does not imply that x_4 is an astronaut.
 From the fact that for all x_4, x_4 is a doctor if and only if x_4 will play squash we can infer that for all x_4, if x_4 plays squash, then x_4 is a doctor via universal biconditional elimination. From the fact that for all x_4, if x_4 were an astronaut, then x_4 is playing squash, and that for all x_4, if x_4 plays squash, then x_4 is a doctor we can infer that for all x_4, if x_4 were an astronaut, then x_4 is a doctor via universal transitivity. From the fact that for all x_4, if x_4 were a doctor, then x_4 is an astronaut, and that for all x_4, if x_4 were an astronaut, then x_4 is a doctor we can infer that for all x_4, x_4 is a doctor if and only if x_4 is an astronaut via universal biconditional introduction. Finally, from the fact that for all x_4, x_4 is a doctor if and only if x_4 is an astronaut we can infer that for all x_4, if x_4 were a doctor, then x_4 is an astronaut via universal biconditional elimination, which contradicts that There is at least one x_4 for which the fact that x_4 were a doctor does not imply that x_4 is an astronaut. Therefore, the answer is no.