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If I am playing a game where I have a card in play that makes it so I don't lose from loss of life, and go to -1 life, would I need to gain 2 life in order to be safe without the anti-loss card anymore, or does the negative 'reset' to 0 after damage effects and cards on the stack resolve?

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You can go into negative life totals. If that doesn't cause you to lose, subsequent events still affect your life total normally: you could gain life back to zero or above, and losing more life will push you further negative.

The rulings on Beacon of Immortality confirm this:

2/1/2007 If you double a negative life total, you do the real math. A life total of -10 becomes -20.

7/15/2007 Beacon of Immortality's effect counts as life gain (or life loss, if the life total was negative) for effects that trigger on or replace life gain (or life loss).

So yes, if you're at -1 life, you'd need to gain 2 life to get back to 1 life and be safe without whatever was keeping you from losing.

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A much more direct example would be Phyrexian Unlife‌​, which on the card actually says "having 0 or less life", and the rulings also describe falling below zero. – Jefromi Dec 8 '15 at 20:36
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You can reach a negative life total even without an effect that prevents you from losing. – Rainbolt Dec 8 '15 at 20:58
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Well, you asked for it. You are playing in a multiplayer game. You control Serra Avatar and Glorious Anthem. Your opponent controls Blood Artist. If you reach 0 life, you lose and Blood Artist does not trigger. If you reach -1 life, you still lose and Blood Artist does trigger. There, I came up with a situation where it matters without one of "those cards". – Rainbolt Dec 8 '15 at 21:24
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I'm all for answers that are correct in every detail when possible, so yes, it's good to avoid implying that you can't go negative if you'd lose. That said, if you look at the body of the question it's pretty clearly asking about a situation that doesn't make much sense if you've lost: going negative, then coming back up. – Jefromi Dec 8 '15 at 21:28
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Since it was just additions, I figured I'd just edit rather than telling you what I thought you should add; feel free to mess with it if you don't like it. – Jefromi Dec 9 '15 at 0:28

You can go to negative life. So if you would go below 0, you keep tracking how much you've lost. If you are at 0 and lose 3, you will be at -3. If you then gain 2, you will be at -1, and would lose the game if you lose your effect that stops you from losing.

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