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Theorem freq1 4127
Description: Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
Assertion
Ref Expression
freq1  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )

Proof of Theorem freq1
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 frforeq1 4126 . . 3  |-  ( R  =  S  ->  (FrFor  R A s  <-> FrFor  S A s ) )
21albidv 1747 . 2  |-  ( R  =  S  ->  ( A. sFrFor  R A s  <->  A. sFrFor  S A s ) )
3 df-frind 4115 . 2  |-  ( R  Fr  A  <->  A. sFrFor  R A s )
4 df-frind 4115 . 2  |-  ( S  Fr  A  <->  A. sFrFor  S A s )
52, 3, 43bitr4g 221 1  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    = wceq 1285  FrFor wfrfor 4110    Fr wfr 4111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-cleq 2076  df-clel 2079  df-ral 2358  df-br 3806  df-frfor 4114  df-frind 4115
This theorem is referenced by:  weeq1  4139
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