Consider the following premises. exists x: ~T(x) or P_4(x). forall x: W(x) -> P_4(x). forall x: T(x) or U(x). forall x: W(x) or ~U(x). forall x: P_4(x) -> W(x). Can we infer exists x: W(x) from them? If possible, name the inference rules being used at each step.
 From forall x: P_4(x) -> W(x), forall x: W(x) -> P_4(x) we can infer forall x: P_4(x) <-> W(x) via universal biconditional introduction. From forall x: W(x) or ~U(x) we can infer forall x: U(x) -> W(x) via universal rewriting. From forall x: T(x) or U(x), exists x: ~T(x) or P_4(x) we can infer exists x: U(x) or P_4(x) via existential resolution. From forall x: P_4(x) <-> W(x) we can infer forall x: P_4(x) -> W(x) via universal biconditional elimination. Finally, from forall x: U(x) -> W(x), forall x: P_4(x) -> W(x), exists x: U(x) or P_4(x) we can infer exists x: W(x) via existential disjunction elimination. Therefore, the answer is yes.