Consider the following premises. For all x_20, if x_20 were a mechanic, then x_20 is a musician. If Joseph were a musician, then Karen is working. If William were an astronaut, then it is sunny. If it is sunny, then Joseph is a mechanic. If Karen works, then William is an astronaut. Can we infer the following from them? If we can, name the inference rule being used: Joseph is a mechanic if and only if Karen is working.
 From the fact that for all x_20, if x_20 were a mechanic, then x_20 is a musician we can infer that if Joseph were a mechanic, then Joseph is a musician via universal instantiation. From the fact that if Karen works, then William is an astronaut, and that if William were an astronaut, then it is sunny we can infer that if Karen works, then it is sunny via transitivity. From the fact that if Joseph were a mechanic, then Joseph is a musician, and that if Joseph were a musician, then Karen is working we can infer that if Joseph were a mechanic, then Karen is working via transitivity. From the fact that if Karen works, then it is sunny, and that if it is sunny, then Joseph is a mechanic we can infer that if Karen works, then Joseph is a mechanic via transitivity. Finally, from the fact that if Joseph were a mechanic, then Karen is working, and that if Karen works, then Joseph is a mechanic we can infer that Joseph is a mechanic if and only if Karen is working via biconditional introduction. Therefore, the answer is yes.