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Theorem freq1 4340
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 4339 . . 3 (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑠 ↔ FrFor 𝑆𝐴𝑠))
21albidv 1824 . 2 (𝑅 = 𝑆 → (∀𝑠 FrFor 𝑅𝐴𝑠 ↔ ∀𝑠 FrFor 𝑆𝐴𝑠))
3 df-frind 4328 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
4 df-frind 4328 . 2 (𝑆 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑆𝐴𝑠)
52, 3, 43bitr4g 223 1 (𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351   = wceq 1353   FrFor wfrfor 4323   Fr wfr 4324
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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-ral 2460  df-br 4001  df-frfor 4327  df-frind 4328
This theorem is referenced by:  weeq1  4352
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