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Theorem caovord2 6251
Description: Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)
Hypotheses
Ref Expression
caovord.1  |-  A  e. 
_V
caovord.2  |-  B  e. 
_V
caovord.3  |-  ( z  e.  S  ->  (
x R y  <->  ( z F x ) R ( z F y ) ) )
caovord2.3  |-  C  e. 
_V
caovord2.com  |-  ( x F y )  =  ( y F x )
Assertion
Ref Expression
caovord2  |-  ( C  e.  S  ->  ( 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    x, F, y, z    x, R, y, z    x, S, y, z

Proof of Theorem caovord2
StepHypRef Expression
1 caovord.1 . . 3  |-  A  e. 
_V
2 caovord.2 . . 3  |-  B  e. 
_V
3 caovord.3 . . 3  |-  ( z  e.  S  ->  (
x R y  <->  ( z F x ) R ( z F y ) ) )
41, 2, 3caovord 6250 . 2  |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
5 caovord2.3 . . . 4  |-  C  e. 
_V
6 caovord2.com . . . 4  |-  ( x F y )  =  ( y F x )
75, 1, 6caovcom 6236 . . 3  |-  ( C F A )  =  ( A F C )
85, 2, 6caovcom 6236 . . 3  |-  ( C F B )  =  ( B F C )
97, 8breq12i 4213 . 2  |-  ( ( C F A ) R ( C F B )  <->  ( A F C ) R ( B F C ) )
104, 9syl6bb 253 1  |-  ( C  e.  S  ->  ( A R B  <->  ( A F C ) R ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   _Vcvv 2948   class class class wbr 4204  (class class class)co 6073
This theorem is referenced by:  caovord3  6252  genpnmax  8876  addclprlem1  8885  mulclprlem  8888  distrlem4pr  8895  ltexprlem6  8910  reclem3pr  8918  ltsosr  8961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076
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