The phrase "brook no denial" is used to convey a sense of uncompromising resolve or determination, and there are several synonyms that can be used interchangeably to convey this sentiment. Some of the most common synonyms for "brook no denial" include "tolerate no refusal," "accept no excuses," "demand compliance," and "insist on obedience." These phrases all convey various shades of authoritative or demanding tones, and can be used to express a wide range of emotions such as urgency, frustration, or even anger. Ultimately, the choice of synonym will depend on the specific context and desired tone of the statement being made.