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Theorem caovord2 6032
Description: Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)
Hypotheses
Ref Expression
caovord.1  |-  A  e. 
_V
caovord.2  |-  B  e. 
_V
caovord.3  |-  ( z  e.  S  ->  (
x R y  <->  ( z F x ) R ( z F y ) ) )
caovord2.3  |-  C  e. 
_V
caovord2.com  |-  ( x F y )  =  ( y F x )
Assertion
Ref Expression
caovord2  |-  ( C  e.  S  ->  ( A R B  <->  ( A F C ) R ( B F C ) ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, F, y, z    x, R, y, z    x, S, y, z

Proof of Theorem caovord2
StepHypRef Expression
1 caovord.1 . . 3  |-  A  e. 
_V
2 caovord.2 . . 3  |-  B  e. 
_V
3 caovord.3 . . 3  |-  ( z  e.  S  ->  (
x R y  <->  ( z F x ) R ( z F y ) ) )
41, 2, 3caovord 6031 . 2  |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
5 caovord2.3 . . . 4  |-  C  e. 
_V
6 caovord2.com . . . 4  |-  ( x F y )  =  ( y F x )
75, 1, 6caovcom 6017 . . 3  |-  ( C F A )  =  ( A F C )
85, 2, 6caovcom 6017 . . 3  |-  ( C F B )  =  ( B F C )
97, 8breq12i 4032 . 2  |-  ( ( C F A ) R ( C F B )  <->  ( A F C ) R ( B F C ) )
104, 9syl6bb 252 1  |-  ( C  e.  S  ->  ( A R B  <->  ( A F C ) R ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023  (class class class)co 5858
This theorem is referenced by:  caovord3  6033  genpnmax  8631  addclprlem1  8640  mulclprlem  8643  distrlem4pr  8650  ltexprlem6  8665  reclem3pr  8673  ltsosr  8716
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-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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