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If I have two promises A and B, only one of which will succeed, how can I get whichever one fulfills successfully? I'm looking for something similar to Promise.race, but which will return only the first promise that fulfills. I'm using promises from ES6.

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Invert the polarity of the promises, and then you can use Promise.all, because it rejects on the first rejected promise, which after inversion corresponds to the first fulfilled promise:

const invert  = p  => new Promise((res, rej) => p.then(rej, res));
const firstOf = ps => invert(Promise.all(ps.map(invert)));

// Utility routines used only in testing.
const wait    = ms => new Promise(res => setTimeout(() => res(ms), ms));
const fail    = f  => Promise.reject(f);
const log     = p  => p.then(v => console.log("pass", v), v => console.log("fail", v));

// Test.
log(firstOf([wait(1000), wait(500) ]));
log(firstOf([wait(1000), fail("f1")]));
log(firstOf([fail("f1"), fail("f2")]));

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