Consider the following premises. forall x_14: P(x_14). forall x_14: Q_14(x_14). exists x_14: P_14(x_14) <-> Q_14(x_14). Can we infer forall x_14: ~(P(x_14) and P_14(x_14)) from them? If possible, name the inference rules being used at each step.
 From exists x_14: P_14(x_14) <-> Q_14(x_14) we can infer exists x_14: Q_14(x_14) -> P_14(x_14) via existential biconditional elimination. From exists x_14: Q_14(x_14) -> P_14(x_14), forall x_14: Q_14(x_14) we can infer exists x_14: P_14(x_14) via existential modus ponens. Finally, from forall x_14: P(x_14), exists x_14: P_14(x_14) we can infer exists x_14: P(x_14) and P_14(x_14) via existential conjunction, which contradicts forall x_14: ~(P(x_14) and P_14(x_14)). Therefore, the answer is no.