If I have Aegis angel on the feild giving my Guardian of the Getless indestructable, and my opponent casts a Day of Judgement is my Guardian of the Gateless destroyed as well? {Aegis Angel} {Guardian of the Gateless} {Day of Judgment}
No. The destroy effect happens to each creature at the same time. When the Guardian gets hit by destroy, it is still protected by Aegis Angel giving it indestructible. Only Aegis Angel would die, then afterward, Guardian will lose indestructible.
It would be different if hit by damage, like {Blasphemous Act}, as that damage would still be on the guardian when the Indestructibility goes away. But because the destroy effect of things like day of judgement happen all at once, it is safe.