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Our goal is a question-answering (QA) system that can show how its answers are implied by its own internal beliefs via a systematic chain of reasoning. Such a capability would allow better understanding of why a model produced the answer it did. Our approach is to recursively combine a trained backward-chaining model, capable of generating a set of premises entailing an answer hypothesis, with a verifier that checks that the model itself believes those premises (and the entailment itself) through self-querying. To our knowledge, this is the first system to generate multistep chains that are both faithful (the answer follows from the reasoning) and truthful (the chain reflects the system's own internal beliefs). In evaluation using two different datasets, users judge that a majority (70%+) of generated chains clearly show how an answer follows from a set of facts - substantially better than a high-performance baseline - while preserving answer accuracy. By materializing model beliefs that systematically support an answer, new opportunities arise for understanding the model's system of belief, and diagnosing and correcting its misunderstandings when an answer is wrong.