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  "body": "\n## Statement\n\nConsider the standard [[Hilbert system|1MGMQX43B8]] for propositional logic, with modus ponens as its only rule.\nLet $\\Gamma$ be a set of formulas, and let $\\varphi$ and $\\psi$ be formulas.\nThen $\\Gamma \\cup \\set{\\varphi} \\vdash \\psi$ if and only if $\\Gamma \\vdash \\varphi \\to \\psi$.\n\n## Remarks\n\nThe direction from right to left is one step of modus ponens.\nFor the other direction, take a derivation $\\psi_1, \\dots, \\psi_n = \\psi$ from $\\Gamma \\cup \\set{\\varphi}$.\nShow by induction on $i$ that $\\Gamma \\vdash \\varphi \\to \\psi_i$.\nIf $\\psi_i$ is an axiom or in $\\Gamma$, use axiom 1, $\\psi_i \\to (\\varphi \\to \\psi_i)$; if $\\psi_i = \\varphi$, use the derivation of $\\varphi \\to \\varphi$.\nIf $\\psi_i$ comes by modus ponens from $\\psi_j$ and $\\psi_j \\to \\psi_i$, use axiom 2 and modus ponens twice.\nFor example, from $\\set{p \\to q, \\; q \\to r, \\; p} \\vdash r$, which takes two steps of modus ponens, the theorem gives $\\set{p \\to q, \\; q \\to r} \\vdash p \\to r$.\nIn first-order logic with the rule of generalization, the theorem needs a hypothesis.\nIt holds when $\\varphi$ is a sentence, or more generally when the derivation never generalizes on a variable free in $\\varphi$.\nWithout this, $\\set{P(x)} \\vdash \\forall x \\, P(x)$ holds, but $P(x) \\to \\forall x \\, P(x)$ is not valid.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that \"assume $\\varphi$ and prove $\\psi$\" is a correct way to prove \"if $\\varphi$ then $\\psi$\".\nMathematicians use this all the time without thinking.\nThe theorem shows that a Hilbert system, which has no rule for assumptions, still allows it.\n\n### L2 human\n\nThe deduction theorem connects the turnstile $\\vdash$ with the connective $\\to$.\nIt is the reason that proofs in a [[Hilbert system|1MGMQX43B8]] can be built from proofs under hypotheses, which keeps them manageable.\nIn [[natural deduction|6KT3BJZ7D1]] the same principle is a primitive rule, implication introduction.\nIts semantic counterpart is that $\\Gamma \\cup \\set{\\varphi} \\models \\psi$ if and only if $\\Gamma \\models \\varphi \\to \\psi$, for [[logical consequence|1HK0VYWTV7]].\n",
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