Consider the following premises. forall y: T_2(y) -> P(y). forall y: W_2(y) <-> P(y). forall y: W_2(y) -> T_2(y). exists y: P(y) -> W_2(y). Can we infer exists y: W_2(y) <-> P(y) from them?
 From forall y: W_2(y) -> T_2(y), forall y: T_2(y) -> P(y) we can infer forall y: W_2(y) -> P(y). From forall y: W_2(y) <-> P(y) we can infer forall y: P(y) -> W_2(y). From forall y: P(y) -> W_2(y), forall y: W_2(y) -> P(y) we can infer forall y: P(y) <-> W_2(y). From forall y: P(y) <-> W_2(y) we can infer forall y: W_2(y) -> P(y). Finally, from forall y: W_2(y) -> P(y), exists y: P(y) -> W_2(y) we can infer exists y: W_2(y) <-> P(y). Therefore, the answer is yes.