Aether Bindings of the Third Age

Instant - Destroy this: Until end of turn, whenever an aura permanent you control with Sigil in its name leaves the arena, amp 1.

Battleworn

Rulings

From the Rosetta + 1st Strike release notes

If two or more aura permanents you control with Sigil in their name leave the arena at the same time, this triggers for each of those auras.
A token aura permanent with Sigil in their name is still considered to have left the arena if it ceases to exist.

Keyword notes

Battleworn - Equipment that wear down after each time they are used to defend. If you defend with a card with Battleworn, put a -1{d} counter on it when the combat chain closes.

When you defend with a card it remains on the combat chain until the chain closes, therefore you cannot defend with an equipment multiple times on the same combat chain.
A -1 counter stays on the card until the card is destroyed or the game ends.
If the defense of a card that is defending is below zero, its defense value is considered to be zero.
When this card is destroyed, it is sent to your graveyard. (Not when it has 0)
When this card is destroyed, it is sent to your graveyard.
If the defense of a card that is defending is below zero, calculate as if its defense is zero.