Suppose there are two statements, $A$ and $B$ that are independent. As far as I know one needn't to prove $A$ or $B$ either, it is enough to generate $C = A \land B$, and then proving $C$ shows $A$ and $B$ are both true. If my understanding is correct, we are practically generalizing $A$ to the extent it embraces $B$ or vice-versa.
Choose, say, $G =$ Goldbach's conjecture, and $R =$ Riemann's conjecture. Suppose they are independent. Suppose someone generates $E = G \land R$. ($E$ would be a nice expression!) Does proving $E$ proves $G$ and proves $R$, too? It seems to be counter-intuitive for me, but - as a layman - I cannot tell you why.
- Are my thoughts about $C$ wrong?
- If not, does the proof on $E$ holds in the terms of proving both $G$ and $R$?
Bonus question: would a solution like this be shocking for the mathematical community for several (at least two) major conjectures?