Full Idea
It might be wondered how one could have any kind of proof procedure at all if transitivity of entailment is disallowed, but the sequent calculus can get around the difficulty.
Gist of Idea
The sequent calculus makes it possible to have proof without transitivity of entailment
Source
John P. Burgess (Philosophical Logic [2009], 5.3)
Book Reference
Burgess,John P.: 'Philosophical Logic' [Princeton 2009], p.105
A Reaction
He gives examples where transitivity of entailment (so that you can build endless chains of deductions) might fail. This is the point of the 'cut free' version of sequent calculus, since the cut rule allows transitivity.