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Theorem frforeq1 4260
 Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
Assertion
Ref Expression
frforeq1 (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))

Proof of Theorem frforeq1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 3926 . . . . . . 7 (𝑅 = 𝑆 → (𝑦𝑅𝑥𝑦𝑆𝑥))
21imbi1d 230 . . . . . 6 (𝑅 = 𝑆 → ((𝑦𝑅𝑥𝑦𝑇) ↔ (𝑦𝑆𝑥𝑦𝑇)))
32ralbidv 2435 . . . . 5 (𝑅 = 𝑆 → (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) ↔ ∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇)))
43imbi1d 230 . . . 4 (𝑅 = 𝑆 → ((∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇)))
54ralbidv 2435 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ ∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇)))
65imbi1d 230 . 2 (𝑅 = 𝑆 → ((∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇) ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇)))
7 df-frfor 4248 . 2 ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
8 df-frfor 4248 . 2 ( FrFor 𝑆𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
96, 7, 83bitr4g 222 1 (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2414   ⊆ wss 3066   class class class wbr 3924   FrFor wfrfor 4244 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2130  df-clel 2133  df-ral 2419  df-br 3925  df-frfor 4248 This theorem is referenced by:  freq1  4261
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