[TYPES] OutsideIn(X) question

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Dear Types List,
I have a small question related to the "OutsideIn(X): Modular type
inference with local assumptions" and was hoping that you could help me a
bit.
My question related about the proof of soundness and principality,
specifically Lemma 7.2 (to be found in page 67). In that lemma, it's stated
that QQ and \phi' Q_q ||- \phi Q_w <-> \phi' Q_w'. I'm trying to recover
the proof (which is omitted in the text), but I stumble upon a wall when
trying to work out what happens in the case an axiom is applied.

In particular, I'm playing with an example where

QQ (the set of axioms) = { forall. C a => D a } (where C and D are
one-parameter type classes)
Q_q = { }
Q_w = { D Int }

Thus, if I apply the rule DINSTW (to be found in page 65), I get a new

Q_w' = { C Int }

Now, if the lemma 7.2 is true, it should be the case that

(1)  QQ ||- C Int <-> D Int

which in particular means that I have the two implications

(2)  { forall. C a => D a, C Int } ||- D Int
(3)  { forall. C a => D a, D Int } ||- C Int

(2) follows easily by applying the AXIOM rule of ||- (as shown in page 54).
However, I don't see how to make (3) work :(
I think that understanding this example will be key for my understanding of
the whole system.

Anybody could point to the error in my reasoning or to places where I could
find more information?
Thanks in advance.