If you are asking for an example of such a method in action, then you have the theorem of Hecke-Deuring-Heilbronn that $h(D) \rightarrow \infty$ as $D \rightarrow \infty$, where $h(D)$ is the class number of the imaginary quadratic field with discriminant $D$.
The Hecke part is that the result is true if the Generalized Riemann hypothesis is true. Deuring-Helbronn part(an exposition here) is that the result is true if the Generalized Riemann hypothesis is false. This is all explained by Dorian Goldfeld in a bulletin article, "Gauss' Class number problem for Imaginary Quadratic Fields".
The proof uses the "Deuring-Heilbronn phenomenon" of repulsion of exceptional zeros of a Dirichlet $L$-function, which is also rather interesting(just saying).