Consider the following premises. forall x_23: P_23(x_23) -> P(x_23). forall x_23: P_10(x_23) -> P(x_23). forall x_23: P(x_23) -> P_23(x_23). P_10(a_10). Can we infer forall x_23: ~P_23(x_23) from them? If possible, name the inference rules being used at each step.
 From forall x_23: P_23(x_23) -> P(x_23), forall x_23: P(x_23) -> P_23(x_23) we can infer forall x_23: P_23(x_23) <-> P(x_23) via universal biconditional introduction. From forall x_23: P_23(x_23) <-> P(x_23) we can infer forall x_23: P(x_23) -> P_23(x_23) via universal biconditional elimination. From P_10(a_10) we can infer exists x_23: P_10(x_23) via existential generalization. From forall x_23: P_10(x_23) -> P(x_23), exists x_23: P_10(x_23) we can infer exists x_23: P(x_23) via existential modus ponens. Finally, from forall x_23: P(x_23) -> P_23(x_23), exists x_23: P(x_23) we can infer exists x_23: P_23(x_23) via existential modus ponens, which contradicts forall x_23: ~P_23(x_23). Therefore, the answer is no.