Consider the following premises. forall y: U(y) -> P(y). forall y: Q(y) -> S_2(y). forall y: Q(y) or ~P(y). forall y: S_2(y) -> Q(y). Can we infer forall y: U(y) -> S_2(y) from them? If possible, name the inference rules being used at each step.
 From forall y: S_2(y) -> Q(y), forall y: Q(y) -> S_2(y) we can infer forall y: S_2(y) <-> Q(y) via universal biconditional introduction. From forall y: S_2(y) <-> Q(y) we can infer forall y: Q(y) -> S_2(y) via universal biconditional elimination. From forall y: Q(y) or ~P(y) we can infer forall y: P(y) -> Q(y) via universal rewriting. From forall y: U(y) -> P(y), forall y: P(y) -> Q(y) we can infer forall y: U(y) -> Q(y) via universal transitivity. Finally, from forall y: U(y) -> Q(y), forall y: Q(y) -> S_2(y) we can infer forall y: U(y) -> S_2(y) via universal transitivity. Therefore, the answer is yes.