It's well-known that given a sentence independent of a system of axioms, we can consistently extend the system by appending either the sentence or its negation. However, if a given theorem can be deduced in each of the two systems, then we have a proof of that theorem. We don't even need to prove independence, because given any sentence, either it or its negation is consistent with a given system, provided that the original axiom system is consistent.
I'm simply wondering whether there's a specific example of that proof technique having been used.