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Theorem caovord2d 5670
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 5669 . 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 5657 . . 3  |-  ( ph  ->  ( C F A )  =  ( A F C ) )
86, 4, 3caovcomd 5657 . . 3  |-  ( ph  ->  ( C F B )  =  ( B F C ) )
97, 8breq12d 3777 . 2  |-  ( ph  ->  ( ( C F A ) R ( C F B )  <-> 
( A F C ) R ( B F C ) ) )
105, 9bitrd 177 1  |-  ( ph  ->  ( A R B  <-> 
( A F C ) R ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   class class class wbr 3764  (class class class)co 5512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515
This theorem is referenced by:  caovord3d  5671  genplt2i  6608  addnqprllem  6625  addnqprulem  6626  mulnqprl  6666  mulnqpru  6667  distrlem4prl  6682  distrlem4pru  6683  1idprl  6688  1idpru  6689  ltexprlemdisj  6704  ltexprlemloc  6705  ltexprlemfl  6707  ltexprlemfu  6709  prplnqu  6718  recexprlem1ssl  6731  recexprlem1ssu  6732  aptiprleml  6737  aptiprlemu  6738  caucvgprlemcanl  6742  cauappcvgprlemlol  6745  cauappcvgprlemloc  6750  cauappcvgprlemladdfu  6752  cauappcvgprlemladdru  6754  cauappcvgprlemladdrl  6755  cauappcvgprlem1  6757  caucvgprlemnkj  6764  caucvgprlemnbj  6765  caucvgprlemlol  6768  caucvgprlemloc  6773  caucvgprlemladdfu  6775  caucvgprlemladdrl  6776  caucvgprprlemnkltj  6787  caucvgprprlemnbj  6791  caucvgprprlemmu  6793  caucvgprprlemlol  6796  caucvgprprlemloc  6801  caucvgprprlemexbt  6804  caucvgprprlemexb  6805  caucvgprprlemaddq  6806  lttrsr  6847  ltsosr  6849  prsrlt  6871  caucvgsrlemoffcau  6882  caucvgsrlemoffgt1  6883  caucvgsrlemoffres  6884  caucvgsr  6886
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