Consider the following premises. forall x: U_3(x) <-> S_1(x). forall x: P(x) or ~U_3(x). forall x: S_1(x) -> U_3(x). Can we infer forall x: S_1(x) -> P(x) from them? If possible, name the inference rules being used at each step.
 From forall x: U_3(x) <-> S_1(x) we can infer forall x: U_3(x) -> S_1(x) via universal biconditional elimination. From forall x: P(x) or ~U_3(x) we can infer forall x: U_3(x) -> P(x) via universal rewriting. From forall x: S_1(x) -> U_3(x), forall x: U_3(x) -> S_1(x) we can infer forall x: S_1(x) <-> U_3(x) via universal biconditional introduction. From forall x: S_1(x) <-> U_3(x) we can infer forall x: S_1(x) -> U_3(x) via universal biconditional elimination. Finally, from forall x: S_1(x) -> U_3(x), forall x: U_3(x) -> P(x) we can infer forall x: S_1(x) -> P(x) via universal transitivity. Therefore, the answer is yes.