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Theorem caovord2 6064
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 6063 . 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 6049 . . 3  |-  ( C F A )  =  ( A F C )
85, 2, 6caovcom 6049 . . 3  |-  ( C F B )  =  ( B F C )
97, 8breq12i 4027 . 2  |-  ( ( C F A ) R ( C F B )  <->  ( A F C ) R ( B F C ) )
104, 9bitrdi 196 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 105    = wceq 1364    e. wcel 2160   _Vcvv 2752   class class class wbr 4018  (class class class)co 5891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5193  df-fv 5239  df-ov 5894
This theorem is referenced by:  caovord3  6065
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