I'm playing against the Dreamer, who has been flipped. A Treacherous Ape (or Illusory Demon) is in play. I do enough damage to kill it.
There are now two effects that seem to be happening "at the same time":
Dreamer:
Whenever a Projection card is destroyed, put it under this card.
TA:
When this card is destroyed, destroy 1 Equipment card.
I see two models of interpreting what happens here that seem reasonable (and I may have missed others)
1) Cards always do their own "on destruction" effects before other "on destruction" effects happen. So the TA destroys an equpment card, then gets put under Dreamer.
2) Since the timing is ambiguous, players choose. If the players choose to trigger Dreamer's effect first, the TA will no longer have any text on it (due to being under Dreamer), so no equipment card will be destroyed.
For the first post I'd go with 1 because it fits what I believe is the spirit of the combo. But I could see it going to players' choice.
Any solid resolution for the first post would cover the second. If you are unsure where to put the card on destruction, choose one. Most players would go for putting it under the Dreamer.
We already know that cards don't get destroyed until they have done all their text reads; we also know that triggered abilities resolve before going on with resolving the triggering card
So I'd read the substitution effect of "put it under the Dreamer instead" as something that happens at the very end of card destruction resolution, meaning TA has time to destroy an equipment before leaving play
"When this card is destroyed" basically pauses the destruction process.
Time responses to destruction like "When a player uses a power." Responses to powers trigger after the power is fully resolved, unlike "When a card is played" which triggers on the card entering play (and vs. one-shots the response effectively goes first)
So responses to destruction should come after the "When this card is destroyed" text resolves.
Cards without that text can have their effects interupted by destruction, but destruction starts, the "When destroyed" text occurs (and cannot be stopped) then after that text resolves responses to the destruction take place.
So TA will destroy an equipment before leaving play, then it leaves play and destruction responses activate.
This process can be wholly circumnavigated by any card that says "when a card would be destroyed do something else instead" Like in Final Wasteland. A card that fits the criteria (destroyed by env. damage for the one in final wasteland) it is not treated as destroyed, and none of the destruction effects will trigger.
Various rulings are referenced here, but putting it together into a sensible order just how I deduce it. Some of those rulings:
1. "When a card enters play" triggers when the card enters play.
1a. Informant cannot have her effect prevented by a one shot that would destroy her.
2. On destruct effects do not trigger when Unforgiving Wasteland text criteria is met, the card is removed from play and not destroyed.
3. "When a Power is used" reaction comes after the power resolves.