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Theorem caovord2 6060
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 6059 . 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 6045 . . 3  |-  ( C F A )  =  ( A F C )
85, 2, 6caovcom 6045 . . 3  |-  ( C F B )  =  ( B F C )
97, 8breq12i 4024 . 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 1363    e. wcel 2158   _Vcvv 2749   class class class wbr 4015  (class class class)co 5888
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891
This theorem is referenced by:  caovord3  6061
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