Consider the following premises. exists x_10: P_5(x_10) <-> P_10(x_10). forall x_10: ~P_5(x_10). forall x_10: P_10(x_10) or ~S_2(x_10). Can we infer forall x_10: S_2(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) or ~S_2(x_10) we can infer forall x_10: S_2(x_10) -> P_10(x_10) via universal rewriting. From exists x_10: P_5(x_10) <-> P_10(x_10), forall x_10: ~P_5(x_10) we can infer exists x_10: ~P_10(x_10) via existential biconditional elimination. Finally, from forall x_10: S_2(x_10) -> P_10(x_10), exists x_10: ~P_10(x_10) we can infer exists x_10: ~S_2(x_10) via existential modus tollens, which contradicts forall x_10: S_2(x_10). Therefore, the answer is no.