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Theorem freq2 4130
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 4129 . . 3  |-  ( A  =  B  ->  (FrFor  R A s  <-> FrFor  R B s ) )
21albidv 1747 . 2  |-  ( A  =  B  ->  ( A. sFrFor  R A s  <->  A. sFrFor  R B s ) )
3 df-frind 4116 . 2  |-  ( R  Fr  A  <->  A. sFrFor  R A s )
4 df-frind 4116 . 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 1283    = wceq 1285  FrFor wfrfor 4111    Fr wfr 4112
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-in 2989  df-ss 2996  df-frfor 4115  df-frind 4116
This theorem is referenced by:  weeq2  4141
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