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Theorem frforeq2 4172
Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
Assertion
Ref Expression
frforeq2 (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))

Proof of Theorem frforeq2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 2562 . . . . 5 (𝐴 = 𝐵 → (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) ↔ ∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇)))
21imbi1d 229 . . . 4 (𝐴 = 𝐵 → ((∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
32raleqbi1dv 2570 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ ∀𝑥𝐵 (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
4 sseq1 3047 . . 3 (𝐴 = 𝐵 → (𝐴𝑇𝐵𝑇))
53, 4imbi12d 232 . 2 (𝐴 = 𝐵 → ((∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇) ↔ (∀𝑥𝐵 (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐵𝑇)))
6 df-frfor 4158 . 2 ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
7 df-frfor 4158 . 2 ( FrFor 𝑅𝐵𝑇 ↔ (∀𝑥𝐵 (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐵𝑇))
85, 6, 73bitr4g 221 1 (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wcel 1438  wral 2359  wss 2999   class class class wbr 3845   FrFor wfrfor 4154
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-in 3005  df-ss 3012  df-frfor 4158
This theorem is referenced by:  freq2  4173
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