I will restrict the answer to indirect proof since that is what concerns the OP.
Wikipedia describes an indirect proof or proof by contradiction as follows:
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by first assuming that the opposite proposition is true, and then shows that such an assumption leads to a contradiction.
The law of the excluded middle is necessary for such a proof to work, that is, either P or ~P is true. An indirect proof negates the proposition we want to prove, derives a contradiction and then claims the proposition itself must be true since by the law of the excluded middle either the proposition or its negation is true.
Some logics do not include the law of the excluded middle such as intuitionistic logic. Wikipedia notes:
Intuitionistic logic can be understood as a weakening of classical logic, meaning that it is more conservative in what it allows a reasoner to infer, while not permitting any new inferences that could not be made under classical logic. Each theorem of intuitionistic logic is a theorem in classical logic, but not conversely.
Many tautologies in classical logic are not theorems in intuitionistic logic - in particular, as said above one of its chief points is to not affirm the law of the excluded middle so as to vitiate the use of non-constructive proof by contradiction which can be used to furnish existence claims without providing explicit examples of the objects that it proves exist.
Hence there exists theorems in classical logic provable using indirect proof or proof by contradiction because of the law of the excluded middle that would not be derivable in this weaker logic.
Wikipedia contributors. (2019, February 6). Intuitionistic logic. In Wikipedia, The Free Encyclopedia. Retrieved 19:29, June 7, 2019, from https://en.wikipedia.org/w/index.php?title=Intuitionistic_logic&oldid=881982480
Wikipedia contributors. (2019, May 6). Proof by contradiction. In Wikipedia, The Free Encyclopedia. Retrieved 19:19, June 7, 2019, from https://en.wikipedia.org/w/index.php?title=Proof_by_contradiction&oldid=895837787