Certainly (2) => (3), at least if you assume function extensionality; it suffices to show that (\Sigma B, A ≃ B) is contractable, and since contractibility and sigma respect equivalence, we can transfer the proof that (\Sigma B, A = B) is contractable. I think the same is not true of (1), though I'm not sure.

On Wed, Jul 19, 2017, 7:26 PM Ian Orton <ri...@cam.ac.uk> wrote:
Consider the following three statements, for all types A and B:

   (1)  (A ≃ B) -> (A = B)
   (2)  (A ≃ B) ≃ (A = B)
   (3)  isEquiv idtoeqv

(3) is the full univalence axiom and we have implications (3) -> (2) ->
(1), but can we say anything about the other directions? Do we have (1)
-> (2) or (2) -> (3)? Can we construct models separating any/all of
these three statements?

Thanks,
Ian

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