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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  A  <->  S  Fr  A ) )

Proof of Theorem freq1
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 frforeq1 4339 . . 3  |-  ( R  =  S  ->  (FrFor  R A s  <-> FrFor  S A s ) )
21albidv 1824 . 2  |-  ( R  =  S  ->  ( A. sFrFor  R A s  <->  A. sFrFor  S A s ) )
3 df-frind 4328 . 2  |-  ( R  Fr  A  <->  A. sFrFor  R A s )
4 df-frind 4328 . 2  |-  ( S  Fr  A  <->  A. sFrFor  S A s )
52, 3, 43bitr4g 223 1  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.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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