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