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Theorem frforeq3 4110
Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
Assertion
Ref Expression
frforeq3 (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))

Proof of Theorem frforeq3
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2143 . . . . . . 7 (𝑆 = 𝑇 → (𝑦𝑆𝑦𝑇))
21imbi2d 228 . . . . . 6 (𝑆 = 𝑇 → ((𝑦𝑅𝑥𝑦𝑆) ↔ (𝑦𝑅𝑥𝑦𝑇)))
32ralbidv 2369 . . . . 5 (𝑆 = 𝑇 → (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) ↔ ∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇)))
4 eleq2 2143 . . . . 5 (𝑆 = 𝑇 → (𝑥𝑆𝑥𝑇))
53, 4imbi12d 232 . . . 4 (𝑆 = 𝑇 → ((∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) ↔ (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
65ralbidv 2369 . . 3 (𝑆 = 𝑇 → (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) ↔ ∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
7 sseq2 3022 . . 3 (𝑆 = 𝑇 → (𝐴𝑆𝐴𝑇))
86, 7imbi12d 232 . 2 (𝑆 = 𝑇 → ((∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) → 𝐴𝑆) ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇)))
9 df-frfor 4094 . 2 ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) → 𝐴𝑆))
10 df-frfor 4094 . 2 ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
118, 9, 103bitr4g 221 1 (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  wral 2349  wss 2974   class class class wbr 3793   FrFor wfrfor 4090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-in 2980  df-ss 2987  df-frfor 4094
This theorem is referenced by:  frind  4115
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