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Theorem caovord2d 6046
Description: Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovordg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  <-> 
( z F x ) R ( z F y ) ) )
caovordd.2  |-  ( ph  ->  A  e.  S )
caovordd.3  |-  ( ph  ->  B  e.  S )
caovordd.4  |-  ( ph  ->  C  e.  S )
caovord2d.com  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
Assertion
Ref Expression
caovord2d  |-  ( ph  ->  ( 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    ph, x, y, z   
x, F, y, z   
x, R, y, z   
x, S, y, z

Proof of Theorem caovord2d
StepHypRef Expression
1 caovordg.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  <-> 
( z F x ) R ( z F y ) ) )
2 caovordd.2 . . 3  |-  ( ph  ->  A  e.  S )
3 caovordd.3 . . 3  |-  ( ph  ->  B  e.  S )
4 caovordd.4 . . 3  |-  ( ph  ->  C  e.  S )
51, 2, 3, 4caovordd 6045 . 2  |-  ( ph  ->  ( A R B  <-> 
( C F A ) R ( C F B ) ) )
6 caovord2d.com . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
76, 4, 2caovcomd 6033 . . 3  |-  ( ph  ->  ( C F A )  =  ( A F C ) )
86, 4, 3caovcomd 6033 . . 3  |-  ( ph  ->  ( C F B )  =  ( B F C ) )
97, 8breq12d 4018 . 2  |-  ( ph  ->  ( ( C F A ) R ( C F B )  <-> 
( A F C ) R ( B F C ) ) )
105, 9bitrd 188 1  |-  ( ph  ->  ( A R B  <-> 
( A F C ) R ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4005  (class class class)co 5877
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880
This theorem is referenced by:  caovord3d  6047  genplt2i  7511  addnqprllem  7528  addnqprulem  7529  mulnqprl  7569  mulnqpru  7570  distrlem4prl  7585  distrlem4pru  7586  1idprl  7591  1idpru  7592  ltexprlemdisj  7607  ltexprlemloc  7608  ltexprlemfl  7610  ltexprlemfu  7612  prplnqu  7621  recexprlem1ssl  7634  recexprlem1ssu  7635  aptiprleml  7640  aptiprlemu  7641  caucvgprlemcanl  7645  cauappcvgprlemlol  7648  cauappcvgprlemloc  7653  cauappcvgprlemladdfu  7655  cauappcvgprlemladdru  7657  cauappcvgprlemladdrl  7658  cauappcvgprlem1  7660  caucvgprlemnkj  7667  caucvgprlemnbj  7668  caucvgprlemlol  7671  caucvgprlemloc  7676  caucvgprlemladdfu  7678  caucvgprlemladdrl  7679  caucvgprprlemnkltj  7690  caucvgprprlemnbj  7694  caucvgprprlemmu  7696  caucvgprprlemlol  7699  caucvgprprlemloc  7704  caucvgprprlemexbt  7707  caucvgprprlemexb  7708  caucvgprprlemaddq  7709  lttrsr  7763  ltsosr  7765  prsrlt  7788  caucvgsrlemoffcau  7799  caucvgsrlemoffgt1  7800  caucvgsrlemoffres  7801  caucvgsr  7803
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