Consider the following premises. forall x_7: ~S_1(x_7). exists x_7: U(x_7) -> P_3(x_7). forall x_7: P_3(x_7) <-> S_1(x_7). Can we infer exists x_7: ~U(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_3(x_7) <-> S_1(x_7) we can infer forall x_7: P_3(x_7) -> S_1(x_7) via universal biconditional elimination. From forall x_7: P_3(x_7) -> S_1(x_7), forall x_7: ~S_1(x_7) we can infer forall x_7: ~P_3(x_7) via universal modus tollens. Finally, from exists x_7: U(x_7) -> P_3(x_7), forall x_7: ~P_3(x_7) we can infer exists x_7: ~U(x_7) via existential modus tollens. Therefore, the answer is yes.