**Theorem.** The Stone-Čech remainder of the real line contains  $2^\mathfrak c$ topologically distinct continua.  

Here a *continuum* is defined to be a compact connected Hausdorff space. $2^\mathfrak c$ is easily seen to be an upper bound in the problem.

The proof was divided into two cases:

Case 1: The Continuum Hypothesis fails. 

<cite authors="Dow, Alan">_Dow, Alan_, [**Some set-theory, Stone-Čech, and $F$-spaces**](http://dx.doi.org/10.1016/j.topol.2011.06.007), Topology Appl. 158, No. 14, 1749-1755 (2011).</cite>

Case 2: The Continuum Hypothesis holds.  

<cite authors="Dow, Alan; Hart, Klaas Pieter">_Dow, Alan; Hart, Klaas Pieter_, [**On subcontinua and continuous images of $\beta \mathbb{R} \setminus \mathbb{R}$**](http://dx.doi.org/10.1016/j.topol.2015.09.017), Topology Appl. 195, 93-106 (2015). </cite>

The proofs are radically different.  In fact, the continua constructed for  Case 1 are all homeomorphic under CH, and CH is essential to the constructions in Case 2.  

This is the only theorem I know of which was proved using CH in this way. 
 What's especially interesting about this proof method is that we know both cases are necessary because CH and $\neg$CH are each consistent with ZFC. This is possibly different from the use of RH and $\neg$RH or some other conjecture and its negation. If the conjecture is eventually proved, for instance, then the other half of your proof was superfluous.