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