Equivalence induction says that in order to prove something for all equivalences, it is enough to prove it for all identity equivalences for all types.

This follows from univalence. But also, conversely, univalence follows from it:

http://www.cs.bham.ac.uk/~mhe/agda-new/UF-Univalence.html#JEq

Is this known? Some years ago it was claimed in this list that equivalence induction would be strictly weaker than univalence.

To prove the above, I apply a technique I learned from Peter Lumsdaine, that given an abstract identity system (Id, refl , J) with no given "computation rule" for J, produces another identity system (Id, refl , J' , J'-comp) with

a "propositional computation rule" J'-comp for J'.

http://www.cs.bham.ac.uk/~mhe/agda-new/Lumsdaine.html

Martin