Is there any proverb that is used ironically to make fun of someone's bravery?
For example, when someone boasts about something they say they have achieved, but in fact, it was already achieved by others.

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And what if we are talking about false bravery? What do you say (sarcastically) when someone acts bravely when there is already no danger?

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that dog is all bark and no bite

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"A legend in his own lunchtime", maybe? Or "He talks a good game." Less personally, there's "Success has many fathers, failure is an orphan."