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I have Angel with the Secret Name in my hand, I put him in battle to make a rescue attempt, with the intention to exchange him. My opponent has Plagued with Diseases up. AwtSN's ability kicks in before the discard from the decrease, right?
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Yes. Calf prevents. PwD is a decrease, which is a different story. AWtSN enters battle, so his ability activates, as it is not prevented/negated. If he's still in battle, then he immediately dies.