Name a theorem T that has a proof based upon the truth of a conjecture C, and also has another proof based upon the falsehood of the same conjecture C, but for longtime has no known direct proof that is independent of C. For instance, it would be a claim that can be proved if P=NP, and can also be proved if P is different from NP. I apologize if the question was previously asked (I also apologize for the title if it does not faithfully reflect the question).

If you are asking for an example of such a method in action, then you have the theorem of HeckeDeuringHeilbronn 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 there are no Siegel zeros of Lfunctions for imaginary quadratic fields. Siegel zeros are exceptional zeros occurring in the real line in the interval $(\frac{1}{2}, 1)$. The DeuringHelbronn part is that the result is true if there are Siegel zeros. The proof uses an effect of "repulsion" of such zeros, which is called the DeuringHeilbronn phenomenon. This is all explained by Dorian Goldfeld in a bulletin article, "Gauss' Class number problem for Imaginary Quadratic Fields". The existence of Siegel zeros is a stronger version of the negation of the Generalized Riemann hypothesis. Hopefully in future the generalized Riemann hypothesis would be proved and thus hopefully it will be shown that the study of Sigel zeros had been just the study of the empty set. Later story(added just for additional information): This method was later strengthened by Landau, Siegel and so on, and finally with more recent developments on the BirchSwinnertonDyer conjecture by Gross and Zagier, an effective version of this theorem was proved by Dorian Goldfeld, and the explicit constants were computed by Joseph Oesterlé. Thus the Gauss class number problem was solved in its entirety. Thanks to Keith Conrad for correcting ambiguities. 


Another standard example is Littlewood's theorem that the number of primes less than x is sometimes greater than Li(x). His proof used different arguments depending on whether the Riemann hypothesis is true or false. See http://en.wikipedia.org/wiki/Skewes%27_number 


How about this basic one: Theorem: There exists an irrational number $p$ and an irrational number $q$ such that $p^q$ is rational. Proof: Let $q=\sqrt{2}$ and note that it is irrational. Conjecture: $q^q$ is irrational. If this conjecture is false, then we are done (the theorem is proved with $p=q$). If the conjecture is true, then let $p=q^q$ and note that it is irrational. The theorem is true in this case as well since $$p^q=(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}}=\sqrt{2}^2=2$$ is rational. 

