If text reads "XX with AA or BB," do both AA and BB have to be on XX?
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Yes, both have to be on XX. For example, if text reads "level 4 or lower Digimon card with the [CS] trait or is yellow or black," it refers to a level 4 or lower Digimon card that has the [CS] trait or a level 4 or lower Digimon card that is yellow or black.
Source: EX12-036 Ryugumon