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Theorem freq1 4275
 Description: Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
Assertion
Ref Expression
freq1 (𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))

Proof of Theorem freq1
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 frforeq1 4274 . . 3 (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑠 ↔ FrFor 𝑆𝐴𝑠))
21albidv 1797 . 2 (𝑅 = 𝑆 → (∀𝑠 FrFor 𝑅𝐴𝑠 ↔ ∀𝑠 FrFor 𝑆𝐴𝑠))
3 df-frind 4263 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
4 df-frind 4263 . 2 (𝑆 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑆𝐴𝑠)
52, 3, 43bitr4g 222 1 (𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330   = wceq 1332   FrFor wfrfor 4258   Fr wfr 4259 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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136  df-ral 2422  df-br 3939  df-frfor 4262  df-frind 4263 This theorem is referenced by:  weeq1  4287
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