Consider the following premises. exists x: Q_3(x) <-> P_3(x). forall x: Q_3(x) <-> P_3(x). Can we infer exists x: P_3(x) -> Q_3(x) from them?
 From exists x: Q_3(x) <-> P_3(x) we can infer exists x: Q_3(x) -> P_3(x). From forall x: Q_3(x) <-> P_3(x) we can infer forall x: P_3(x) -> Q_3(x). From forall x: P_3(x) -> Q_3(x), exists x: Q_3(x) -> P_3(x) we can infer exists x: P_3(x) <-> Q_3(x). Finally, from exists x: P_3(x) <-> Q_3(x) we can infer exists x: P_3(x) -> Q_3(x). Therefore, the answer is yes.