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Theorem caovord2d 5698
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 5697 . 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 5685 . . 3  |-  ( ph  ->  ( C F A )  =  ( A F C ) )
86, 4, 3caovcomd 5685 . . 3  |-  ( ph  ->  ( C F B )  =  ( B F C ) )
97, 8breq12d 3805 . 2  |-  ( ph  ->  ( ( C F A ) R ( C F B )  <-> 
( A F C ) R ( B F C ) ) )
105, 9bitrd 181 1  |-  ( ph  ->  ( A R B  <-> 
( A F C ) R ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   class class class wbr 3792  (class class class)co 5540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by:  caovord3d  5699  genplt2i  6666  addnqprllem  6683  addnqprulem  6684  mulnqprl  6724  mulnqpru  6725  distrlem4prl  6740  distrlem4pru  6741  1idprl  6746  1idpru  6747  ltexprlemdisj  6762  ltexprlemloc  6763  ltexprlemfl  6765  ltexprlemfu  6767  prplnqu  6776  recexprlem1ssl  6789  recexprlem1ssu  6790  aptiprleml  6795  aptiprlemu  6796  caucvgprlemcanl  6800  cauappcvgprlemlol  6803  cauappcvgprlemloc  6808  cauappcvgprlemladdfu  6810  cauappcvgprlemladdru  6812  cauappcvgprlemladdrl  6813  cauappcvgprlem1  6815  caucvgprlemnkj  6822  caucvgprlemnbj  6823  caucvgprlemlol  6826  caucvgprlemloc  6831  caucvgprlemladdfu  6833  caucvgprlemladdrl  6834  caucvgprprlemnkltj  6845  caucvgprprlemnbj  6849  caucvgprprlemmu  6851  caucvgprprlemlol  6854  caucvgprprlemloc  6859  caucvgprprlemexbt  6862  caucvgprprlemexb  6863  caucvgprprlemaddq  6864  lttrsr  6905  ltsosr  6907  prsrlt  6929  caucvgsrlemoffcau  6940  caucvgsrlemoffgt1  6941  caucvgsrlemoffres  6942  caucvgsr  6944
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