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Theorem freq1 4169
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 4168 . . 3  |-  ( R  =  S  ->  (FrFor  R A s  <-> FrFor  S A s ) )
21albidv 1752 . 2  |-  ( R  =  S  ->  ( A. sFrFor  R A s  <->  A. sFrFor  S A s ) )
3 df-frind 4157 . 2  |-  ( R  Fr  A  <->  A. sFrFor  R A s )
4 df-frind 4157 . 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 1287    = wceq 1289  FrFor wfrfor 4152    Fr wfr 4153
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-cleq 2081  df-clel 2084  df-ral 2364  df-br 3844  df-frfor 4156  df-frind 4157
This theorem is referenced by:  weeq1  4181
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