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Theorem caovord2d 6047
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 6046 . 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 6034 . . 3  |-  ( ph  ->  ( C F A )  =  ( A F C ) )
86, 4, 3caovcomd 6034 . . 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 5878
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 5881
This theorem is referenced by:  caovord3d  6048  genplt2i  7512  addnqprllem  7529  addnqprulem  7530  mulnqprl  7570  mulnqpru  7571  distrlem4prl  7586  distrlem4pru  7587  1idprl  7592  1idpru  7593  ltexprlemdisj  7608  ltexprlemloc  7609  ltexprlemfl  7611  ltexprlemfu  7613  prplnqu  7622  recexprlem1ssl  7635  recexprlem1ssu  7636  aptiprleml  7641  aptiprlemu  7642  caucvgprlemcanl  7646  cauappcvgprlemlol  7649  cauappcvgprlemloc  7654  cauappcvgprlemladdfu  7656  cauappcvgprlemladdru  7658  cauappcvgprlemladdrl  7659  cauappcvgprlem1  7661  caucvgprlemnkj  7668  caucvgprlemnbj  7669  caucvgprlemlol  7672  caucvgprlemloc  7677  caucvgprlemladdfu  7679  caucvgprlemladdrl  7680  caucvgprprlemnkltj  7691  caucvgprprlemnbj  7695  caucvgprprlemmu  7697  caucvgprprlemlol  7700  caucvgprprlemloc  7705  caucvgprprlemexbt  7708  caucvgprprlemexb  7709  caucvgprprlemaddq  7710  lttrsr  7764  ltsosr  7766  prsrlt  7789  caucvgsrlemoffcau  7800  caucvgsrlemoffgt1  7801  caucvgsrlemoffres  7802  caucvgsr  7804
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