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Theorem frforeq2 4341
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 2672 . . . . 5 (𝐴 = 𝐵 → (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) ↔ ∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇)))
21imbi1d 231 . . . 4 (𝐴 = 𝐵 → ((∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
32raleqbi1dv 2680 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ ∀𝑥𝐵 (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇)))
4 sseq1 3178 . . 3 (𝐴 = 𝐵 → (𝐴𝑇𝐵𝑇))
53, 4imbi12d 234 . 2 (𝐴 = 𝐵 → ((∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇) ↔ (∀𝑥𝐵 (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐵𝑇)))
6 df-frfor 4327 . 2 ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
7 df-frfor 4327 . 2 ( FrFor 𝑅𝐵𝑇 ↔ (∀𝑥𝐵 (∀𝑦𝐵 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐵𝑇))
85, 6, 73bitr4g 223 1 (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  wral 2455  wss 3129   class class class wbr 4000   FrFor wfrfor 4323
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-in 3135  df-ss 3142  df-frfor 4327
This theorem is referenced by:  freq2  4342
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