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  "body": "\n## Summary\n\nEquational logic is the fragment of first-order logic whose statements are [[identities|PKD440G2XW]] between terms and whose proofs use only equational reasoning.\nIts rules are reflexivity, symmetry, transitivity, replacement of equals inside an operation, and [[substitution|NCDZ969V7X]] of terms for variables.\nBy Birkhoff's completeness theorem these rules derive exactly the identities that follow semantically from a given set of identities.\nIt is the logic of algebraic manipulation, and it underlies term rewriting and automated equational reasoning.\n\n## Explanations\n\n### L1 human\n\nEquational logic is the logic of high school algebra.\nYou start from some laws, such as $x + 0 \\approx x$ and $x + y \\approx y + x$.\nYou may plug any expressions into a law, replace equals by equals inside a bigger expression, and chain equalities together.\nFor example, from these two laws one derives $0 + x \\approx x + 0$ by substituting into commutativity, and then $0 + x \\approx x$ by transitivity.\n",
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