Does the phrase have official term? I'd be glad to know, since that the "to be" is in an odd position.
and we usually find this scenario
Q: They like the kittens. A: So does everyone.
Q: She is pretty. A: So is her sister.
But how about if the sentences contain modal verbs.
Q: We may lose the event. A: So does everyone, so may everyone or so is everyone?.
Q: We could lose the event. A: So did everyone, so could everyone or so was everyone?.
The question is also for the rest of modal verbs, such as will, shall, might, would, should.