CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Semantic equivalence is preserved by substitution

Statement

If $ϕ$ and $ψ$ have the same truth at every assignment in every $L$ -structure, then so do $ϕ ⟨ t / x ⟩$ and $ψ ⟨ t / x ⟩$ . Moreover, in any fixed structure and assignment, terms $t, u$ of equal value give the same truth for $ϕ ⟨ t / x ⟩$ and $ϕ ⟨ u / x ⟩$ .

Facts & Assumptions

Given: Work in ZF unless the statement explicitly weakens or supplements it; fix the objects and hypotheses of the statement.

[F1]

There is a canonical total capture-avoiding substitution $ϕ ⟨ t / x ⟩$ on formulas, obtained by renaming obstructing binders using least unused variable indices. It replaces the original free occurrences of $x$ and satisfies $M, s ⊨ ϕ ⟨ t / x ⟩ ⟺ M, s [x : = ⟦ t ⟧_{s}] ⊨ ϕ .$ Canonicity refers to the specified coding and traversal, not to literal invariance under other fresh-variable conventions. (Canonical capture-avoiding substitution)

Proof

1.1

Fix a structure and assignment $s$ and put $b = ⟦ t ⟧_{s}$ . The two substituted formulas have the truth values of $ϕ$ and $ψ$ at the same assignment $s [x : = b]$ . The hypothesis applies at this assignment, so they agree.

2.1

If $⟦ t ⟧_{s} = ⟦ u ⟧_{s}$ , the updated assignments are literally equal. Apply the substitution identity once for each term to obtain equality of the two truth values.

F1∎

Depends on

Canonical capture-avoiding substitution

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Moschovakis, Lecture Notes in Logic (2014) — x1.3.3 p.48; Weiss chapter 1 Equality Principle p.18. (standard reference, not scraped)