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  "body": "\n## Statement\n\nLet $\\sigma$ be a signature with only function symbols and let $X$ be an infinite set of variables.\nLet $\\Sigma$ be a set of [[identities|PKD440G2XW]] over $\\sigma$ and $X$.\nFor every identity $s \\approx t$ over $\\sigma$ and $X$, the following are equivalent:\n\n1. $\\Sigma \\vdash s \\approx t$, that is, $s \\approx t$ is derivable from $\\Sigma$ in [[equational logic|72S3YSQFZ7]];\n2. $\\Sigma \\models s \\approx t$, that is, every algebra of signature $\\sigma$ that satisfies all identities of $\\Sigma$ satisfies $s \\approx t$.\n\n## Remarks\n\nSoundness, from 1 to 2, holds because each rule of equational logic preserves truth in every algebra.\nFor completeness, let $\\theta$ be the set of pairs $(u, v)$ of terms with $\\Sigma \\vdash u \\approx v$.\nThe rules say that $\\theta$ is a congruence of the term algebra that is invariant under substitutions, and the quotient of the term algebra by $\\theta$ satisfies $\\Sigma$.\nEvaluating $s$ and $t$ in this quotient at the assignment $x \\mapsto x/\\theta$ gives $s/\\theta$ and $t/\\theta$, so if $\\Sigma \\models s \\approx t$ then $(s, t) \\in \\theta$.\nThe quotient is the free algebra on $X$ of the variety defined by $\\Sigma$.\n\nConsequence: a set of identities is an [[equational theory|F1M51B4MDG]] exactly when it contains all identities $t \\approx t$ and is closed under the rules of symmetry, transitivity, replacement and substitution.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that equational reasoning is enough.\nIf a law follows from some given laws in every possible algebra, then it can be proved from them by plugging in, replacing equals by equals, and chaining equalities.\nNothing more powerful, such as case analysis or quantifier tricks, is ever needed for this kind of conclusion.\n\n### L2 human\n\nThis is the equational analogue of Gödel's completeness theorem, with a much simpler set of rules.\nThe proof builds a canonical model: the term algebra divided by provable equality, which is the free algebra of the variety defined by $\\Sigma$.\nTogether with Birkhoff's HSP theorem, it sets up the correspondence between varieties of algebras and equational theories.\nIt also justifies term rewriting: a rewriting proof of $s \\approx t$ is a derivation in equational logic, and every true consequence has such a derivation, though finding it may be undecidable.\n",
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