Consider the following premises. forall x: P(x) <-> Q_4(x). exists x: Q_4(x) <-> P(x). Can we infer exists x: P(x) <-> Q_4(x) from them? If possible, name the inference rules being used at each step.
 From exists x: Q_4(x) <-> P(x) we can infer exists x: Q_4(x) -> P(x) via existential biconditional elimination. From forall x: P(x) <-> Q_4(x) we can infer forall x: P(x) -> Q_4(x) via universal biconditional elimination. Finally, from forall x: P(x) -> Q_4(x), exists x: Q_4(x) -> P(x) we can infer exists x: P(x) <-> Q_4(x) via existential biconditional introduction. Therefore, the answer is yes.