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Theorem freq2 4496
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
StepHypRef Expression
1 eqimss2 3346 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 frss 4492 . . 3  |-  ( B 
C_  A  ->  ( R  Fr  A  ->  R  Fr  B ) )
31, 2syl 16 . 2  |-  ( A  =  B  ->  ( R  Fr  A  ->  R  Fr  B ) )
4 eqimss 3345 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 frss 4492 . . 3  |-  ( A 
C_  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
64, 5syl 16 . 2  |-  ( A  =  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
73, 6impbid 184 1  |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    C_ wss 3265    Fr wfr 4481
This theorem is referenced by:  weeq2  4514  frsn  4890  f1oweALT  6015  frfi  7290  freq12d  26806  ifr0  27323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-in 3272  df-ss 3279  df-fr 4484
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