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My opponent has Athreos, God of Passage on the field. They then play Martyr of Sands and use her ability "{1}, Reveal X white cards from your hand, Sacrifice Martyr of Sands: You gain three times X life." They sacrifice the Martyr, triggering Athreos's ability. I choose not to pay 3 life, and they return the Martyr to their hand.

Does my opponent still gain life even though Martyr of Sands isn't in the graveyard as her ability resolves?

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Your opponent does still gain the life. Nothing in the ability says that the Martyr has to be in their graveyard to gain the life. The sacrifice cost was paid, and once the ability was on the stack, it existed independent of the creature card (CR 112.7a).

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Since nothing but the sacrifice tries to move the Martyr, the Martyr would still be on the battlefield if it wasn't sacrificed. The Martyr was sacrificed, so the cost was payed, so your opponent gains the life[1]. Athreos caused a triggered ability to happen to Martyr after Martyr's ability was put on the stack, but that doesn't matter.

The steps taken in order are:

  1. The costs are paid for Martyr of Sands's ability: One mana is paid, you reveal some number of white cards from your hand, and Martyr of Sands is placed in the graveyard.
  2. Martyr of Sands's ability is placed on the stack, but doesn't resolve yet.
  3. Athreos's triggered ability is triggered and then placed on the stack.
  4. Athreos's ability resolves. In this case, you decide not to pay three life, and Martyr of Sands is returned to your opponent's hand.
  5. Martyr of Sands's ability resolves, causing your opponent to gain three times X life.

Additionally, your opponent would still gain the life even if the sacrifice itself was replaced with another action[CR 117.11][2]: the cost to activate the ability would have changed, but the ability would still have the same effect when resolving.

117.11. The actions performed when paying a cost may be modified by effects. Even if they are, meaning the actions that are performed don’t match the actions that are called for, the cost has still been paid.

  1. The sacrifice happened (even if the creature didn't end up in the graveyard), so abilities that trigger when you sacrifice a creature (such as Dragon Appeasement's second ability) will trigger.
  2. The sacrifice itself was replaced, so abilities that trigger when you sacrifice a creature (such as Dragon Appeasement's second ability) won't trigger[CR 614.6].
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  • Do you want to expand on this using rule 117.11? (addresses a different question, but could be informative for OP)
    – Hao Ye
    Dec 8, 2014 at 22:08
  • @Hao Ye, Good idea, done.
    – ikegami
    Dec 8, 2014 at 22:11
  • Are you saying if for instance i cast magma spray in response to martyr's ability the sacrifice would then be replaced and no life gain would happen?
    – Justin Fincher
    Dec 8, 2014 at 22:57
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    @JustinFincher Sacrificing the Martyr is part of the cost of activating the ability (activated abilities are of the form "cost: effect"). You pay the cost before it goes on the stack and players get priority, so if you activate the ability, once anyone gets a chance to cast a spell, it's already sacrificed.
    – Cascabel
    Dec 8, 2014 at 23:13
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    @ikegami Sure. I do think your point (3) is a little iffy, though. Replacing costs is not hypothetical, but replacing sacrifices is, and that's what makes citing 117.11 confusing in the context of your answer.
    – Cascabel
    Dec 9, 2014 at 1:04
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They will still gain life. This is because sacrificing the Martyr is sacrificed and once that happens the ability doesn't care what happens to her.

What happens is your opponent activates the Martyr's ability, sacrificing her, paying the mana, and revealing white cards from your hand. Once they've done all of that the ability goes on the stack. After it is on the stack Aethreos's ability triggers and when it resolves you allow the Martyr to return to their hand. At this point the Martyr's ability is still on the stack waiting to resolve, and since there is nothing in the effect that requires the Martyr to be in the graveyard when it resolves it is perfectly happy to give your opponent life.

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