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Theorem freq2 4823
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 3496 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 frss 4819 . . 3  |-  ( B 
C_  A  ->  ( R  Fr  A  ->  R  Fr  B ) )
31, 2syl 17 . 2  |-  ( A  =  B  ->  ( R  Fr  A  ->  R  Fr  B ) )
4 eqimss 3495 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 frss 4819 . . 3  |-  ( A 
C_  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
64, 5syl 17 . 2  |-  ( A  =  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
73, 6impbid 195 1  |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1454    C_ wss 3415    Fr wfr 4808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-in 3422  df-ss 3429  df-fr 4811
This theorem is referenced by:  weeq2  4841  frsn  4923  f1oweALT  6803  frfi  7841  freq12d  35941  ifr0  36846
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