Consider the following premises. forall x: W(x) -> P(x). forall x: T(x) -> P(x). forall x: U(x) -> W(x). forall x: T_4(x) or U(x). forall x: T_4(x) -> T(x). Can we infer forall x: P(x) from them? If possible, name the inference rules being used at each step.
 From forall x: U(x) -> W(x), forall x: W(x) -> P(x) we can infer forall x: U(x) -> P(x) via universal transitivity. From forall x: T_4(x) -> T(x), forall x: T(x) -> P(x) we can infer forall x: T_4(x) -> P(x) via universal transitivity. Finally, from forall x: T_4(x) -> P(x), forall x: U(x) -> P(x), forall x: T_4(x) or U(x) we can infer forall x: P(x) via universal disjunction elimination. Therefore, the answer is yes.