Ex. If I capture a Carvanha in the Dream World that has the hidden ability Speed Boost. Will it also have the regular ability of Rough Skin? Or are they totally separate?
A pokemon only has one ability. Caravanha will only have it's dream world ability, unless it doesn't have a dream world ability like Hydreigon will only have levitate, Hydreigon has no dream world ability.
Hope this helps :)
