Here is one direct example:
The whole notion of NP-completeness came about when Cook and Levin and others discovered that it was possible to express the existence of an accepting path of computation in a non-deterministic Turing machine in a fairly straightforward manner as a Boolean expression as long as the length of the computation could be bounded.
So all of NP was reduced to the satisfiability of a certain Boolean expression that could easily be constructed from a hypothetical non-deterministic Turing machine whose running time could be bounded in a polynomial in the size of the input.
But the problem of the satisfiability of a Boolean expression is trivially in NP itself: simply assign all the variables non-determistically, evaluate the expression, and accept if it is true. There will be an accepting path of computation if and only if the expression is satisfiable. Therefore Boolean satisfiability is said to be NP-complete.
Likewise, for all existing problems known to be NP-complete, there is an explicit reduction that could be applied to solve any problem in NP in deterministic polynomical time if a suitable algorithm were found for that problem.
But the possibility of finding a polynomial algorithm for a problem known only indirectly to be NP-complete (without an explicit reduction) has not been ruled out, and this would be the case for any non-constructive proof that P=NP, since if P=NP then all problems in P are trivially NP-complete.