The Condition Holds vs is Satisfied vs Obtains

Solution 1:

• If a condition holds, it still exists
• If a condition is satisfied, then all necessary features or qualities are present
• If a condition obtains, it "exists, is used, or is accepted"

Saying that a condition obtains is more formal and academic, and is marked as such in the dictionary reference cited.

Saying that a condition is satisfied may or may not imply that it wasn't previously.

Saying that a condition holds implies its continued persistence.

However, in the context of trying to say that the condition is true, and continues to be so, all of these terms are more or less interchangeable. I wouldn't vary them simply for the sake of avoiding repetition; I would revise my writing so it wasn't necessary to assert that the condition is true so often.

Solution 2:

For what it's worth, I would say that in my experience, the two most common usages are that the condition (a) "is satisfied" or (b) "is met," which are more or less interchangable. Less often, I've heard people say that the condition "holds." I've never heard anyone say that a condition "obtains," and I think that I would find that usage understandable but a bit confusing.