Barendregt's proof of subject reduction for $\lambda2$
I still think there is an imprecision in how he uses the lemma. However there is a solution (I must thank to Barbara Petit who came with the solution).

In fact, the solution comes from the definition of $\geq$ (def. 4.2.1), which is morally this:

$\sigma&gt;\rho\quad$ if $\quad\dfrac{\Gamma\vdash P:\sigma}{\Gamma\vdash P:\rho}$

However, instead of defining it in that way, he defines the relation in terms of the types only. The advantage on defining it in terms of sequents, is that we can deduce that if $\sigma&gt;\forall\alpha.\sigma$, then $\alpha\notin FV(\Gamma)$, which is exactly what he needs in the proof (and from where the imprecision comes).