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Theorem caovord2d 6223
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 6222 . 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 6210 . . 3  |-  ( ph  ->  ( C F A )  =  ( A F C ) )
86, 4, 3caovcomd 6210 . . 3  |-  ( ph  ->  ( C F B )  =  ( B F C ) )
97, 8breq12d 4193 . 2  |-  ( ph  ->  ( ( C F A ) R ( C F B )  <-> 
( A F C ) R ( B F C ) ) )
105, 9bitrd 245 1  |-  ( ph  ->  ( A R B  <-> 
( A F C ) R ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180  (class class class)co 6048
This theorem is referenced by:  caovord3d  6224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051
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