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If there are no Archfiends on your side of the field, "Falling Down" is destroyed immediately, even in the middle of a chain.
The Archfiend you need for "Falling Down" doesn't have to be a monster. Cards like "Axe of Despair", or an Archfiend monster equipped to your "Relinquished", will prevent "Falling Down" from being destroyed.
You can activate "Falling Down" when there are no Archfiends on your side of the field, but it is destroyed immediately before it is equipped to a monster.
You cannot use "Barrel Behind the Door" against the damage from "Falling Down" because you cannot use "Barrel Behind the Door" against a Continuous or Equip Spell Card.
If "Falling Down" takes control of an Archfi end monster, and the controlling player's only other Archfiend card is removed from the field, "Falling Down" is not destroyed by its effect.
"Falling Down" may be activated as long as there is an Archfi end on the controller's field. This includes non-monster Archfiends.