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Theorem frforeq2 4227
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 2600 . . . . 5 (𝐴 = 𝐵 → (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) ↔ ∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇)))
21imbi1d 230 . . . 4 (𝐴 = 𝐵 → ((∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
32raleqbi1dv 2608 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ ∀𝑥𝐵 (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
4 sseq1 3086 . . 3 (𝐴 = 𝐵 → (𝐴𝑇𝐵𝑇))
53, 4imbi12d 233 . 2 (𝐴 = 𝐵 → ((∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇) ↔ (∀𝑥𝐵 (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐵𝑇)))
6 df-frfor 4213 . 2 ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
7 df-frfor 4213 . 2 ( FrFor 𝑅𝐵𝑇 ↔ (∀𝑥𝐵 (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐵𝑇))
85, 6, 73bitr4g 222 1 (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wcel 1463  wral 2390  wss 3037   class class class wbr 3895   FrFor wfrfor 4209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-in 3043  df-ss 3050  df-frfor 4213
This theorem is referenced by:  freq2  4228
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