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Theorem frforeq3 4084
 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 2101 . . . . . . 7 (𝑆 = 𝑇 → (𝑦𝑆𝑦𝑇))
21imbi2d 219 . . . . . 6 (𝑆 = 𝑇 → ((𝑦𝑅𝑥𝑦𝑆) ↔ (𝑦𝑅𝑥𝑦𝑇)))
32ralbidv 2326 . . . . 5 (𝑆 = 𝑇 → (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) ↔ ∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇)))
4 eleq2 2101 . . . . 5 (𝑆 = 𝑇 → (𝑥𝑆𝑥𝑇))
53, 4imbi12d 223 . . . 4 (𝑆 = 𝑇 → ((∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) ↔ (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
65ralbidv 2326 . . 3 (𝑆 = 𝑇 → (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) ↔ ∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
7 sseq2 2967 . . 3 (𝑆 = 𝑇 → (𝐴𝑆𝐴𝑇))
86, 7imbi12d 223 . 2 (𝑆 = 𝑇 → ((∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) → 𝐴𝑆) ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇)))
9 df-frfor 4068 . 2 ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) → 𝐴𝑆))
10 df-frfor 4068 . 2 ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
118, 9, 103bitr4g 212 1 (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ⊆ wss 2917   class class class wbr 3764   FrFor wfrfor 4064 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-in 2924  df-ss 2931  df-frfor 4068 This theorem is referenced by:  frind  4089
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