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  "body": "\n## Statement\n\nLet $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.\nIn classical logic, the negation of $\\forall x \\in X, \\, P(x)$ is logically equivalent to $\\exists x \\in X, \\, \\neg P(x)$.\n$$\\neg \\bigl( \\forall x \\in X, \\, P(x) \\bigr) \\equiv \\exists x \\in X, \\, \\neg P(x)$$\n\n## Remarks\n\nThe direction from $\\exists x, \\, \\neg P(x)$ to $\\neg \\forall x, \\, P(x)$ is valid in intuitionistic logic. The other direction needs classical logic.\nThis law explains why one [[counterexample|0ED72CTBNP]] refutes a universal statement.\nFor a finite $X$, it is a form of [[De Morgan's laws|09PKJKA6J0]].\n\n## Explanations\n\n### L1 human\n\n\"Not every student passed\" means \"some student did not pass\".\nTo deny that all elements have a property, you show one element without it.\n",
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