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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature.\nThe set $\\operatorname{FV}(\\varphi)$ of *free variables* of a [[formula|HYJ10AF6RT]] $\\varphi$ over $\\sigma$ is defined by recursion on $\\varphi$:\n\n1. for an atomic formula, $\\operatorname{FV}(\\varphi)$ is the set of variables that occur in its [[terms|6D3WA7N2W4]];\n2. $\\operatorname{FV}(\\neg \\varphi) = \\operatorname{FV}(\\varphi)$, and $\\operatorname{FV}(\\varphi \\ast \\psi) = \\operatorname{FV}(\\varphi) \\cup \\operatorname{FV}(\\psi)$ for each binary connective $\\ast$;\n3. $\\operatorname{FV}(\\forall x \\, \\varphi) = \\operatorname{FV}(\\exists x \\, \\varphi) = \\operatorname{FV}(\\varphi) \\setminus \\set{x}$.\n\nA *sentence* over $\\sigma$ is a formula $\\varphi$ with $\\operatorname{FV}(\\varphi) = \\emptyset$.\n\n## Remarks\n\nThe formula $\\forall x \\, \\exists y \\, (x \u003c y)$ is a sentence.\nIndeed $\\operatorname{FV}(x \u003c y) = \\set{x, y}$, removing $y$ gives $\\set{x}$, and removing $x$ gives $\\emptyset$.\nThe formula $\\exists y \\, (x \u003c y)$ is not a sentence, since $x$ remains free.\nSentences are also called closed formulas.\nIn Mathlib sentences are `FirstOrder.Language.Sentence`.\n",
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