Consider the following premises. forall x: T_4(x) -> W_1(x). exists x: U(x) or ~W_1(x). forall x: U(x) -> T_4(x). forall x: T_4(x) -> U(x). Can we infer exists x: W_1(x) <-> U(x) from them? If possible, name the inference rules being used at each step.
 From exists x: U(x) or ~W_1(x) we can infer exists x: W_1(x) -> U(x) via existential rewriting. From forall x: U(x) -> T_4(x), forall x: T_4(x) -> U(x) we can infer forall x: U(x) <-> T_4(x) via universal biconditional introduction. From forall x: U(x) <-> T_4(x) we can infer forall x: U(x) -> T_4(x) via universal biconditional elimination. From forall x: U(x) -> T_4(x), forall x: T_4(x) -> W_1(x) we can infer forall x: U(x) -> W_1(x) via universal transitivity. Finally, from exists x: W_1(x) -> U(x), forall x: U(x) -> W_1(x) we can infer exists x: W_1(x) <-> U(x) via existential biconditional introduction. Therefore, the answer is yes.