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Better yet, discard an angel from hand for a double instead.
While you are correct that Chamber will take precedence generally due to the ruling you gave, Antiochus must first complete before it triggers, when we get down to a really-technical what-activates-when, because Chamber is a triggered ability. Thus, the holder of the angel may use the instead on Antiochus, which then means that the discard never took place and Chamber never triggers as a result.You may discard instead before it goes to Chamber.
What is the difference between the Antiochus and Wandering Spirit instead abilities that has Antiochus take precedence over Chamber but Tartaros take precedence over Wandering Spirit? Is it that Wandering Spirit's ability is also a trigegred one?