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Fix the particular occurrence of $\alpha$ that is being replaced. The subformula analysis above makes it a node of the formula syntax tree of $\varphi$. Proceed by structural induction on the surrounding logical formula.

If this node is the root, the subformula replacement gives $\beta$, already a logical formula. Otherwise, if $\varphi =\neg \psi$, the chosen occurrence lies inside $\psi$. The induction hypothesis gives a logical formula ${\psi }^{\prime }$ after the subformula replacement there, and the whole result is the logical formula $\neg {\psi }^{\prime }$. If $\varphi =(\psi \to \chi )$, a proper chosen occurrence lies in exactly one of the two children. Replacing inside that child gives either $({\psi }^{\prime }\to \chi )$ or $(\psi \to {\chi }^{\prime })$, each a logical formula by the formation rule. A logical variable has no proper subformula, so there is no additional base case.

Therefore replacing the designated subformula occurrence by any logical formula preserves well-formedness. The argument concerns one chosen occurrence; it does not require simultaneous subformula replacement of every occurrence of the same string.

Fix any Boolean valuation $v$. Logical equivalence of $\alpha$ and $\beta$ means $v(\alpha )=v(\beta )$ for this Boolean valuation and every other one. Follow the unique path from the replaced node to the root of the formula syntax tree. At a negation node, equal input truth values have equal negations. At a logical implication node, the unaffected child has the same truth value before and after subformula replacement, while the affected child has the same truth value by induction; the resulting logical implications therefore agree as well.

Induction along that finite path yields $v(\varphi )=v({\varphi }^{\prime })$. Since $v$ was arbitrary,This is the replacement theorem for logical equivalence: the semantics of each logical connective depends only on the truth values of its immediate inputs.

Subformula replacement is the operation of replacing one specified subformula occurrence by another logical formula. Its preservation of well-formedness is the subformula replacement theorem.