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Lethal damage, damage from a source with deathtouch, and effects that say “destroy” won’t cause a permanent with indestructible to be put into the graveyard. However, a permanent with indestructible can be put into the graveyard for a number of reasons. The most likely reasons are if it’s sacrificed, if it’s legendary and another legendary permanent with the same name is controlled by the same player, if it’s a planeswalker and another planeswalker with the same subtype is controlled by the same player, if it’s a creature with toughness 0 or less, or if it’s an Aura that’s either unattached or attached to something illegal.

If a creature enchanted by Indestructibility is dealt lethal damage, the creature isn’t destroyed, but the damage remains marked on the creature. If Indestructibility stops enchanting that creature later in the turn, the creature will no longer have indestructible and will be destroyed.

If an effect would simultaneously destroy Indestructibility and the creature it’s enchanting, only Indestructibility is destroyed.