I am studying predicate calculus on some lecture notes on my own. I have a question concerning a strange rule of inference called the Thinning Rule which is stated from the writer as the third rule of inference for the the formal system K$(L)$ (after Modus Ponens and the Generalisation Rule):
TR) $ $ if $\Gamma \vdash \phi$ and $\Gamma \subset \Delta$, then $\Delta \vdash \phi$.
Well, it seems to me that TR is not necessary at all since it is easily proven from the very definition of formal proof (without TR, of course). I am not able to see what is the point here.
The Notes are here http://www.maths.ox.ac.uk/system/files/coursematerial/2011/2369/4/logic.pdf (page 14-15)