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Theorem freq2 4171
Description: Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
freq2  |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )

Proof of Theorem freq2
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 frforeq2 4170 . . 3  |-  ( A  =  B  ->  (FrFor  R A s  <-> FrFor  R B s ) )
21albidv 1752 . 2  |-  ( A  =  B  ->  ( A. sFrFor  R A s  <->  A. sFrFor  R B s ) )
3 df-frind 4157 . 2  |-  ( R  Fr  A  <->  A. sFrFor  R A s )
4 df-frind 4157 . 2  |-  ( R  Fr  B  <->  A. sFrFor  R B s )
52, 3, 43bitr4g 221 1  |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-in 3005  df-ss 3012  df-frfor 4156  df-frind 4157
This theorem is referenced by:  weeq2  4182
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