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Theorem caovord2d 6156
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 6155 . 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 6143 . . 3  |-  ( ph  ->  ( C F A )  =  ( A F C ) )
86, 4, 3caovcomd 6143 . . 3  |-  ( ph  ->  ( C F B )  =  ( B F C ) )
97, 8breq12d 4138 . 2  |-  ( ph  ->  ( ( C F A ) R ( C F B )  <-> 
( A F C ) R ( B F C ) ) )
105, 9bitrd 244 1  |-  ( ph  ->  ( A R B  <-> 
( A F C ) R ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   class class class wbr 4125  (class class class)co 5981
This theorem is referenced by:  caovord3d  6157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-ov 5984
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