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Theorem freq2 4345
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 4344 . . 3  |-  ( A  =  B  ->  (FrFor  R A s  <-> FrFor  R B s ) )
21albidv 1824 . 2  |-  ( A  =  B  ->  ( A. sFrFor  R A s  <->  A. sFrFor  R B s ) )
3 df-frind 4331 . 2  |-  ( R  Fr  A  <->  A. sFrFor  R A s )
4 df-frind 4331 . 2  |-  ( R  Fr  B  <->  A. sFrFor  R B s )
52, 3, 43bitr4g 223 1  |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351    = wceq 1353  FrFor wfrfor 4326    Fr wfr 4327
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-in 3135  df-ss 3142  df-frfor 4330  df-frind 4331
This theorem is referenced by:  weeq2  4356
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