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Theorem frforeq3 4309
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 2221 . . . . . . 7 (𝑆 = 𝑇 → (𝑦𝑆𝑦𝑇))
21imbi2d 229 . . . . . 6 (𝑆 = 𝑇 → ((𝑦𝑅𝑥𝑦𝑆) ↔ (𝑦𝑅𝑥𝑦𝑇)))
32ralbidv 2457 . . . . 5 (𝑆 = 𝑇 → (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) ↔ ∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇)))
4 eleq2 2221 . . . . 5 (𝑆 = 𝑇 → (𝑥𝑆𝑥𝑇))
53, 4imbi12d 233 . . . 4 (𝑆 = 𝑇 → ((∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) ↔ (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
65ralbidv 2457 . . 3 (𝑆 = 𝑇 → (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) ↔ ∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
7 sseq2 3152 . . 3 (𝑆 = 𝑇 → (𝐴𝑆𝐴𝑇))
86, 7imbi12d 233 . 2 (𝑆 = 𝑇 → ((∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) → 𝐴𝑆) ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇)))
9 df-frfor 4293 . 2 ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) → 𝐴𝑆))
10 df-frfor 4293 . 2 ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
118, 9, 103bitr4g 222 1 (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1335  wcel 2128  wral 2435  wss 3102   class class class wbr 3967   FrFor wfrfor 4289
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ral 2440  df-in 3108  df-ss 3115  df-frfor 4293
This theorem is referenced by:  frind  4314
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