Sure. If you have a substitution principle in your logic that allows you to substitute identical expressions for each other in any sentence following the identification, you don't need to posit the transitivity of identity as an axiom; you can just deduce it as follows:

1. Suppose: a = b & b = c
2. | a = b (&-Elim, 1)
3. | b = c (&-Elim, 1)
4. | a = c (=-Elim, 2-3)
5. (a = b & b = c) => a =c (direct proof, 1-4)

Hope that's the sort of thing you were looking for.