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Theorem freq2 4545
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 3393 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 frss 4541 . . 3  |-  ( B 
C_  A  ->  ( R  Fr  A  ->  R  Fr  B ) )
31, 2syl 16 . 2  |-  ( A  =  B  ->  ( R  Fr  A  ->  R  Fr  B ) )
4 eqimss 3392 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 frss 4541 . . 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 1652    C_ wss 3312    Fr wfr 4530
This theorem is referenced by:  weeq2  4563  frsn  4940  f1oweALT  6066  frfi  7344  freq12d  27104  ifr0  27620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-in 3319  df-ss 3326  df-fr 4533
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