The phrase "to establish that belief" means to provide evidence or facts that make a particular idea or opinion seem true or valid to others. Full definition
There are certain established beliefs about modern British elections. (theconversation.com)
Portrait of an Artist is a project inspired by seven remarkable female artists that drive culture by shaking the structures of established belief systems. (artefuse.com)
You can not grow without being challenged or processing through information that isn't cohesive with your already established beliefs. (religion.blogs.cnn.com)