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