Consider the following premises. forall y: W_2(y) <-> P_2(y). exists y: P_2(y) <-> P(y). Can we infer exists y: P(y) -> W_2(y) from them? If possible, name the inference rules being used at each step.
 From forall y: W_2(y) <-> P_2(y) we can infer forall y: P_2(y) -> W_2(y) via universal biconditional elimination. From exists y: P_2(y) <-> P(y) we can infer exists y: P(y) -> P_2(y) via existential biconditional elimination. Finally, from exists y: P(y) -> P_2(y), forall y: P_2(y) -> W_2(y) we can infer exists y: P(y) -> W_2(y) via existential transitivity. Therefore, the answer is yes.