"this idea holds" is correct and can be used in written English.
You can use it when you want to express that something is true or correct. For example: "There is a common belief that people who work hard will be successful; this idea holds true in most cases."
This idea held that, in the absence of a legitimate authority, the people had the right to appoint their own leaders.
(This idea held sway for decades, until recent research found that bullies had relatively higher self-esteem than their victims, and that most bullies had, themselves, been bullied. Today we understand these categorical personality types more as performances of various continua).
This idea held sway for many years, before it too was revealed to be an oversimplification of how genetic information is transferred [ 3].
Contradiction or not, the idea holds water.
The modern immune system, the idea holds, is stymied by the sudden absence of its customary microbial targets.
This intriguing idea holds water less because of the conclusion Meyer reaches - that Holmes was traumatised by the discovery that his mother had committed adultery with Professor Moriarty - than the fact that the methods of the father of psychoanalysis and the father of modern detective fiction are so closely allied.
This popular idea holds that huge schools of fish and shellfish beds permitted early fishers and foragers to settle along the coast, build elaborate architecture, and develop complex societies, moving inland only later.
This controversial idea holds that each time a memory is recalled it becomes briefly vulnerable--and can be weakened by compounds that target certain genes or molecules in the brain.
This same idea holds true for diets and nutrition, business and entrepreneurship, writing and art, and virtually any other endeavor we attempt.
