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Theorem freq1 4234
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 4233 . . 3 (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑠 ↔ FrFor 𝑆𝐴𝑠))
21albidv 1778 . 2 (𝑅 = 𝑆 → (∀𝑠 FrFor 𝑅𝐴𝑠 ↔ ∀𝑠 FrFor 𝑆𝐴𝑠))
3 df-frind 4222 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
4 df-frind 4222 . 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 1312   = wceq 1314   FrFor wfrfor 4217   Fr wfr 4218
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-cleq 2108  df-clel 2111  df-ral 2396  df-br 3898  df-frfor 4221  df-frind 4222
This theorem is referenced by:  weeq1  4246
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