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Elesh Norn, Mother of Machines

Vigilance
If a permanent entering causes a triggered ability of a permanent you control to trigger, that ability triggers an additional time.
Permanents entering don't cause abilities of permanents your opponents control to trigger.

Phyrexia: All Will Be One

Vigilance
If a permanent entering the battlefield causes a triggered ability of a permanent you control to trigger, that ability triggers an additional time.
Permanents entering the battlefield don't cause abilities of permanents your opponents control to trigger.

Abilities that apply "as [this permanent] enters the battlefield," such as choosing a card type with Stenn, Paranoid Partisan, are also unaffected.

Elesh Norn's abilities don't look at who controls the permanent entering the battlefield, only who controls the permanent that has the triggered ability. For example, if you control Elesh Norn and a permanent with the ability "Whenever a creature enters the battlefield, you gain 1 life," a creature entering the battlefield under an opponent's control will cause that ability to trigger twice.

Elesh Norn's first ability doesn't copy the triggered ability; it just causes the ability to trigger twice. Any choices made as you put the ability onto the stack, such as modes and targets, are made separately for each instance of the ability. Any choices made on resolution, such as whether to put counters on a permanent, are also made individually.

Elesh Norn's second and third ability each affects a permanent's own enters-the-battlefield triggered abilities as well as other triggered abilities that trigger when that permanent enters the battlefield. Such triggered abilities start with "when" or "whenever." Some keyword abilities also include a triggered ability that happens when a permanent enters the battlefield, such as For Mirrodin!

If a permanent entering the battlefield at the same time as Elesh Norn (including Elesh Norn itself) causes a triggered ability of a permanent you control to trigger, that ability triggers an additional time.

If a triggered ability is linked to a second ability, additional instances of that triggered ability are also linked to that second ability. If the second ability refers to "the exiled card," it refers to all cards exiled by instances of the triggered ability.

If you somehow control two Elesh Norns, a permanent entering the battlefield causes abilities to trigger three times, not four. A third Elesh Norn causes abilities to trigger four times, a fourth causes abilities to trigger five times, and so on.

In a Two-Headed Giant game, Elesh Norn does not affect triggered abilities of permanents your teammate controls.

In some cases involving linked abilities, an ability requires information about "the exiled card." When this happens, the ability gets multiple answers. If these answers are being used to determine the value of a variable, the sum is used. For example, if Elite Arcanist's enters-the-battlefield ability triggers twice, two cards are exiled. The value of X in the activation cost of Elite Arcanist's other ability is the sum of the two cards' mana values. As the ability resolves, you create copies of both cards and can cast none, one, or both of the copies in any order.

Replacement effects are unaffected by Elesh Norn's abilities. For example, a creature that enters the battlefield under your control with one +1/+1 counter on it won't receive an additional +1/+1 counter.

Similarly, a permanent entering the battlefield at the same time as Elesh Norn (still including Elesh Norn) won't cause abilities of permanents an opponent controls to trigger.