Consider the following premises. forall x_10: U(x_10) or S_1(x_10). forall x_10: P(x_10) -> U(x_10). forall x_10: T_4(x_10) -> T(x_10). forall x_10: ~T(x_10). forall x_10: U(x_10) -> P(x_10). forall x_10: P(x_10) <-> T_4(x_10). Can we infer exists x_10: ~S_1(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: T_4(x_10) -> T(x_10), forall x_10: ~T(x_10) we can infer forall x_10: ~T_4(x_10) via universal modus tollens. From forall x_10: U(x_10) -> P(x_10), forall x_10: P(x_10) -> U(x_10) we can infer forall x_10: U(x_10) <-> P(x_10) via universal biconditional introduction. From forall x_10: P(x_10) <-> T_4(x_10), forall x_10: ~T_4(x_10) we can infer forall x_10: ~P(x_10) via universal biconditional elimination. From forall x_10: U(x_10) <-> P(x_10), forall x_10: ~P(x_10) we can infer forall x_10: ~U(x_10) via universal biconditional elimination. Finally, from forall x_10: U(x_10) or S_1(x_10), forall x_10: ~U(x_10) we can infer forall x_10: S_1(x_10) via universal disjunctive syllogism, which contradicts exists x_10: ~S_1(x_10). Therefore, the answer is no.