“that it applies to everyone equally,” is not a property of the rule of law”
The term “rule of law” has a very specific meaning. Google the definition.
If laws don’t apply equally then by definition that system of laws is not a “rule of law” system. In other words the term doesn’t refer to just any sets of laws.
If you are going to stick to that definition, then “rule of law” is not one of the essential characteristics of Western civilization. It was not a founding principle of the US, since slaves were not equal under the law.