You cant kill him before he casts hellfire as he will hellfire at 15-20% no matter how long he has left on the nail phase timer, meaning he can cast hellfire 30 seconds into the nail phase if you have enough damage on him instead of the usual 1 minute.
Also eruptions shouldn't be a problem if the dps are moving correctly, in phase 1 you can move and continue damaging ifrit with no downtime, in phase 3 you should have very little to no downtime depending on if it spawns on you and phase 5 with 6 lancers should be dead before he has a chance to use more than 1-2 abilties.
Either way congrats on the kill guys very nice, although 6 minutes isn't really that fast from personal experience but your healer was very impressive.