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Theorem freq12d 27113
Description: Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
weeq12d.l  |-  ( ph  ->  R  =  S )
weeq12d.r  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
freq12d  |-  ( ph  ->  ( R  Fr  A  <->  S  Fr  B ) )

Proof of Theorem freq12d
StepHypRef Expression
1 weeq12d.l . . 3  |-  ( ph  ->  R  =  S )
2 freq1 4552 . . 3  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
31, 2syl 16 . 2  |-  ( ph  ->  ( R  Fr  A  <->  S  Fr  A ) )
4 weeq12d.r . . 3  |-  ( ph  ->  A  =  B )
5 freq2 4553 . . 3  |-  ( A  =  B  ->  ( S  Fr  A  <->  S  Fr  B ) )
64, 5syl 16 . 2  |-  ( ph  ->  ( S  Fr  A  <->  S  Fr  B ) )
73, 6bitrd 245 1  |-  ( ph  ->  ( R  Fr  A  <->  S  Fr  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    Fr wfr 4538
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-ral 2710  df-rex 2711  df-in 3327  df-ss 3334  df-br 4213  df-fr 4541
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