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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature.\nA first-order [[formula|HYJ10AF6RT]] over $\\sigma$ is in *prenex normal form* if it has the form\n$$Q_1 x_1 \\, Q_2 x_2 \\cdots Q_n x_n \\, \\psi,$$\nwhere $n \\ge 0$ and each $Q_i$ is $\\forall$ or $\\exists$.\nThe $x_i$ are variables.\nHere $\\psi$ is a formula with no quantifiers.\nThe string $Q_1 x_1 \\cdots Q_n x_n$ is the *prefix* and $\\psi$ is the *matrix*.\n\n## Remarks\n\nThe formula $\\forall x \\, \\exists y \\, (R(x, y) \\lor \\neg P(x))$ is in prenex normal form: its prefix is $\\forall x \\, \\exists y$ and its matrix has no quantifiers.\nThe formula $\\neg \\forall x \\, P(x)$ is not: the quantifier is inside a negation, so the formula is not a prefix followed by a quantifier-free matrix.\nIt is equivalent to $\\exists x \\, \\neg P(x)$, which is in prenex normal form.\nIn classical first-order logic, every formula is logically equivalent to a formula in prenex normal form.\nThe proof moves quantifiers outwards with rules such as $(\\forall x \\, \\varphi) \\to \\psi \\equiv \\exists x \\, (\\varphi \\to \\psi)$, when $x$ is not free in $\\psi$.\nThis rule uses that structures have nonempty domains.\n",
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