In API testing, deriving logical constraints on API response bodies to be used as oracles is crucial for generating test cases and performing automated testing of RESTful APIs. However, existing approaches are restricted to dynamic analysis, in which oracles are extracted via the execution of APIs as part of the system under test. In this paper, we propose a complementary LLM-based static approach in which constraints for API response bodies are mined from API specifications. We leverage large language models (LLMs) to comprehend API specifications, mine constraints for response bodies, and generate test cases. To reduce LLM hallucination, we apply an Observation-Confirmation (OC) scheme that uses initial prompts to contextualize constraints, allowing subsequent prompts to more accurately confirm their presence. Our empirical results show that RBCTest with OC prompting achieves high precision in constraint mining, with averages ranging from 85.1% to 93.6%. It also performs well in generating test cases from mined constraints, with precision ranging from 86.4% to 91.7%. We further use test cases generated by RBCTest to detect 46 mismatches between API specifications and actual response data across 19 real-world APIs. Four of these mismatches were reported in developers' forums.