The growing excitement around the ability of large language models (LLMs) to tackle various tasks has been tempered by their propensity for generating unsubstantiated information (hallucination) and by their inability to effectively handle inconsistent inputs. To detect such issues, we propose the novel task of Query-Conditioned Natural Language Inference (QC-NLI), where the goal is to determine the semantic relationship (e.g. entailment or not entailment) between two documents conditioned on a query; we demonstrate that many common tasks regarding inconsistency detection can be formulated as QC-NLI problems. We focus on three applications in particular: fact verification, intrinsic hallucination detection, and document inconsistency detection. We convert existing datasets for these tasks into the QC-NLI format, and manual annotation confirms their high quality. Finally, we employ zero- and few-shot prompting methods to solve the QC-NLI prediction problem for each task, showing the critical importance of conditioning on the query.