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Theorem freq1 4236
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 4235 . . 3  |-  ( R  =  S  ->  (FrFor  R A s  <-> FrFor  S A s ) )
21albidv 1780 . 2  |-  ( R  =  S  ->  ( A. sFrFor  R A s  <->  A. sFrFor  S A s ) )
3 df-frind 4224 . 2  |-  ( R  Fr  A  <->  A. sFrFor  R A s )
4 df-frind 4224 . 2  |-  ( S  Fr  A  <->  A. sFrFor  S A s )
52, 3, 43bitr4g 222 1  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1314    = wceq 1316  FrFor wfrfor 4219    Fr wfr 4220
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-cleq 2110  df-clel 2113  df-ral 2398  df-br 3900  df-frfor 4223  df-frind 4224
This theorem is referenced by:  weeq1  4248
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