Consider the following premises. forall x: P_3(x) -> R(x). forall x: S(x) <-> R(x). exists x: P_3(x) <-> S(x). Can we infer exists x: P_3(x) -> S(x) from them? If possible, name the inference rules being used at each step.
 From forall x: S(x) <-> R(x) we can infer forall x: R(x) -> S(x) via universal biconditional elimination. From forall x: P_3(x) -> R(x), forall x: R(x) -> S(x) we can infer forall x: P_3(x) -> S(x) via universal transitivity. From exists x: P_3(x) <-> S(x) we can infer exists x: S(x) -> P_3(x) via existential biconditional elimination. From exists x: S(x) -> P_3(x), forall x: P_3(x) -> S(x) we can infer exists x: S(x) <-> P_3(x) via existential biconditional introduction. Finally, from exists x: S(x) <-> P_3(x) we can infer exists x: P_3(x) -> S(x) via existential biconditional elimination. Therefore, the answer is yes.