When should one include the proof of known results in a mathematical PhD thesis? When writing a PhD thesis on mathematics one needs to quote many results by others, such as

The following theorem was proved by ABC in [1].
Theorem. Bla, bla, bla...

My question is when should one include a proof of the result in his/her PhD thesis, if it is basically the same as in the reference? I do not quite see the point of copy-pasting the proof by others. Of course if one has a completely different proof of the same result, it is probably suitable to include it.
As a general rule, you can cite other people's theorems without explaining their proofs, and omitting a proof is a good idea if it would be a lengthy distraction.  However, there are several reasons why including such a proof could be helpful:


Including it may be convenient for the reader if the proof is short.  It's annoying to look up another paper and discover that you only needed a short argument that could easily have been explained in the original paper.  Extracting information from a reference can be cumbersome (you have to locate exactly what you're looking for, figure out what it depends on, sort out the notation, etc.), while giving your own explanation can help readers avoid some of these difficulties.
Even if the proof is not particularly short, it may serve as a warm-up for new applications of the same techniques.  Reminding the reader how they work may make your paper much easier to read than if you just dive into the newest and most complicated case.


Ph.D. dissertations are something of a special case, because your advisor may encourage you to include extra details in the background sections (beyond what you might include in a published paper).  This is partly a matter of demonstrating your mastery of the area and partly a matter of writing a useful survey for others.  Advisors differ in how they approach this: some think it's a waste of time and it's best just to focus on writing a published paper, while others think writing a more extensive dissertation is a valuable learning exercise.  This is an issue you should discuss with your advisor.