I know the title may not be fully descriptive, but please bear with me.
When I do math, whether I have formed a conjecture and seek to prove it or am simply proving nontrivial theorems for homework assignments, I will in some cases look for "intermediate" steps, so to speak. For example, if I know that $A \Rightarrow B$, $B \Rightarrow C$, and $C \Rightarrow D$, then proving $A \Rightarrow D$ simply amounts to wisely choosing these intermediate steps to get the result.
What I am interested in is the opposite. For example, if a mathematician was attempting to prove $A \Rightarrow D$, and he/she also knew that $A \Rightarrow B$ and $C \Rightarrow D$, that person may attempt to prove that $B \Rightarrow C$ to complete the proof. Are there any historical examples of a situation like this where $B \Rightarrow C$ had never previously been rigorously proven, but was eventually shown to be true as a result of someone's attempt to prove $A \Rightarrow D$?
I apologize for the soft question, but I was just thinking about how I do math one day, and in almost every case I can think of, I am using these "smaller" results as building blocks to a larger proof, which seems generally logical. Even in textbooks, almost every result is presented this way because it makes the most sense for students understand the pieces of a proof before they can understand the whole thing. However, it would not surprise me if researchers grappling with open questions have taken the approach I described here, so I am curious if there are any noteworthy results that were discovered this way.