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  "body": "\n## Statement\n\nLet $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.\nThe negation of $\\exists x \\in X, \\, P(x)$ is logically equivalent to $\\forall x \\in X, \\, \\neg P(x)$.\n$$\\neg \\bigl( \\exists x \\in X, \\, P(x) \\bigr) \\equiv \\forall x \\in X, \\, \\neg P(x)$$\n\n## Remarks\n\nThis equivalence is valid in intuitionistic logic as well as in classical logic.\nTogether with the [[negation of a universal statement|0P3A06QNDP]], it gives a rule. To negate a quantified statement, swap each $\\forall$ and $\\exists$ and negate the predicate.\n\n## Explanations\n\n### L1 human\n\n\"No student failed\" means \"every student did not fail\".\nTo deny that some element has a property, you must show that each element lacks it.\n",
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