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Theorem frforeq1 4364
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 4023 . . . . . . 7 (𝑅 = 𝑆 → (𝑦𝑅𝑥𝑦𝑆𝑥))
21imbi1d 231 . . . . . 6 (𝑅 = 𝑆 → ((𝑦𝑅𝑥𝑦𝑇) ↔ (𝑦𝑆𝑥𝑦𝑇)))
32ralbidv 2490 . . . . 5 (𝑅 = 𝑆 → (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) ↔ ∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇)))
43imbi1d 231 . . . 4 (𝑅 = 𝑆 → ((∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇)))
54ralbidv 2490 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ ∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇)))
65imbi1d 231 . 2 (𝑅 = 𝑆 → ((∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇) ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇)))
7 df-frfor 4352 . 2 ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
8 df-frfor 4352 . 2 ( FrFor 𝑆𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
96, 7, 83bitr4g 223 1 (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  wcel 2160  wral 2468  wss 3144   class class class wbr 4021   FrFor wfrfor 4348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-cleq 2182  df-clel 2185  df-ral 2473  df-br 4022  df-frfor 4352
This theorem is referenced by:  freq1  4365
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