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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature with only function symbols and let $X$ be a set of variables.\nLet $\\Sigma$ be a set of [[identities|PKD440G2XW]] over $\\sigma$ and $X$.\nThe identities *derivable* from $\\Sigma$ in *equational logic*, written $\\Sigma \\vdash s \\approx t$, form the smallest set of identities that satisfies the following rules, for all [[terms|6D3WA7N2W4]] $s, t, u, s_i, t_i$:\n\n1. (axioms) every identity in $\\Sigma$ is derivable;\n2. (reflexivity) $t \\approx t$ is derivable;\n3. (symmetry) if $s \\approx t$ is derivable, so is $t \\approx s$;\n4. (transitivity) if $s \\approx t$ and $t \\approx u$ are derivable, so is $s \\approx u$;\n5. (replacement) if $s_i \\approx t_i$ is derivable for $i = 1, \\ldots, n$ and $f$ is a function symbol of arity $n$, then $f(s_1, \\ldots, s_n) \\approx f(t_1, \\ldots, t_n)$ is derivable;\n6. (substitution) if $s \\approx t$ is derivable and $\\theta$ is a [[substitution|NCDZ969V7X]], then $s\\theta \\approx t\\theta$ is derivable.\n\n## Remarks\n\nA derivation is a finite list of identities, each one in $\\Sigma$ or obtained from earlier ones by a rule.\nEach rule preserves truth in every algebra, so every derivable identity holds in every algebra that satisfies $\\Sigma$; Birkhoff's completeness theorem gives the converse.\n\nExample: let $\\Sigma = \\set{x + 0 \\approx x, \\ x + y \\approx y + x}$.\nRule 6 applied to the second axiom with $\\theta = \\set{x \\mapsto 0, \\ y \\mapsto x}$ gives $0 + x \\approx x + 0$, and rule 4 with the first axiom gives $\\Sigma \\vdash 0 + x \\approx x$.\n\nNon-example: $x + y \\approx y$ is not derivable from $\\Sigma$.\nThe group $(\\Z, +, 0)$ satisfies $\\Sigma$, and every derivable identity holds in it, but $1 + 0 \\ne 0$.\n",
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