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Theorem freq2 4364
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 3231 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 frss 4360 . . 3  |-  ( B 
C_  A  ->  ( R  Fr  A  ->  R  Fr  B ) )
31, 2syl 15 . 2  |-  ( A  =  B  ->  ( R  Fr  A  ->  R  Fr  B ) )
4 eqimss 3230 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 frss 4360 . . 3  |-  ( A 
C_  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
64, 5syl 15 . 2  |-  ( A  =  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
73, 6impbid 183 1  |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    C_ wss 3152    Fr wfr 4349
This theorem is referenced by:  weeq2  4382  frsn  4760  f1oweALT  5851  frfi  7102  freq12d  27135  ifr0  27653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-fr 4352
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