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  "body": "\n## Summary\n\nIn a statement with several quantifiers, the order of different quantifiers changes the meaning.\nFor a [[predicate|0T3DPBG981]] $P(x, y)$, the statement $\\exists y \\, \\forall x \\, P(x, y)$ implies $\\forall x \\, \\exists y \\, P(x, y)$, but the converse is false in general.\nTwo quantifiers of the same type commute: $\\forall x \\, \\forall y \\, P(x, y)$ is equivalent to $\\forall y \\, \\forall x \\, P(x, y)$, and the same holds for $\\exists$.\n\n## Remarks\n\nOn the integers, $\\forall x \\, \\exists y \\, (y \u003e x)$ is true, but $\\exists y \\, \\forall x \\, (y \u003e x)$ is false.\nIn $\\forall x \\, \\exists y$, the choice of $y$ can depend on $x$.\nThe definitions of continuity and of uniform continuity differ only in the order of quantifiers.\n\n## Explanations\n\n### L1 human\n\n\"Every person has a mother\" is $\\forall x \\, \\exists y$: each person has their own mother.\n\"Some person is the mother of every person\" is $\\exists y \\, \\forall x$: one mother for all.\nThe first sentence is true, and the second is false.\n",
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