Consider the following premises. forall x_10: P_7(x_10) -> ~Q_7(x_10). forall x_10: P_7(x_10) <-> P_2(x_10). forall x_10: R_10(x_10) -> P_2(x_10). forall x_10: P_2(x_10) -> Q_10(x_10). forall x_10: Q_10(x_10) -> R_10(x_10). forall x_10: Q_7(x_10). Can we infer exists x_10: R_10(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_2(x_10) -> Q_10(x_10), forall x_10: Q_10(x_10) -> R_10(x_10) we can infer forall x_10: P_2(x_10) -> R_10(x_10) via universal transitivity. From forall x_10: P_7(x_10) -> ~Q_7(x_10), forall x_10: Q_7(x_10) we can infer forall x_10: ~P_7(x_10) via universal modus tollens. From forall x_10: P_2(x_10) -> R_10(x_10), forall x_10: R_10(x_10) -> P_2(x_10) we can infer forall x_10: P_2(x_10) <-> R_10(x_10) via universal biconditional introduction. From forall x_10: P_7(x_10) <-> P_2(x_10), forall x_10: ~P_7(x_10) we can infer forall x_10: ~P_2(x_10) via universal biconditional elimination. Finally, from forall x_10: P_2(x_10) <-> R_10(x_10), forall x_10: ~P_2(x_10) we can infer forall x_10: ~R_10(x_10) via universal biconditional elimination, which contradicts exists x_10: R_10(x_10). Therefore, the answer is no.