I activated this card's [On Play] [When Digivolving] effect, and when my opponent's Digimon would leave the battle area for the 1st process, an immediate-type effect such as a "when [...] would leave" effect caused this card to be removed from the battle area. Can I then process the part of the effect after "then" in this card's effect?
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Yes, you can. If an effect activates, it is to be fully resolved even if the card that activated the effect is removed from that area during the processing.
Source: BT20-011 ExVeemon