Consider the following premises. forall x: Q(x) or ~P_3(x). forall x: P_3(x) -> Q(x). exists x: Q(x) -> P_3(x). Can we infer forall x: ~(Q(x) <-> P_3(x)) from them? If possible, name the inference rules being used at each step.
 From exists x: Q(x) -> P_3(x), forall x: P_3(x) -> Q(x) we can infer exists x: Q(x) <-> P_3(x) via existential biconditional introduction. From forall x: Q(x) or ~P_3(x) we can infer forall x: P_3(x) -> Q(x) via universal rewriting. From exists x: Q(x) <-> P_3(x) we can infer exists x: Q(x) -> P_3(x) via existential biconditional elimination. Finally, from exists x: Q(x) -> P_3(x), forall x: P_3(x) -> Q(x) we can infer exists x: Q(x) <-> P_3(x) via existential biconditional introduction, which contradicts forall x: ~(Q(x) <-> P_3(x)). Therefore, the answer is no.