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Theorem caovord3 6262
Description: Ordering law. (Contributed by NM, 29-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 )
caovord3.4  |-  D  e. 
_V
Assertion
Ref Expression
caovord3  |-  ( ( ( B  e.  S  /\  C  e.  S
)  /\  ( A F B )  =  ( C F D ) )  ->  ( A R C  <->  D R B ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, D, y, z    x, F, y, z    x, R, y, z    x, S, y, z

Proof of Theorem caovord3
StepHypRef Expression
1 caovord.1 . . . . 5  |-  A  e. 
_V
2 caovord2.3 . . . . 5  |-  C  e. 
_V
3 caovord.3 . . . . 5  |-  ( z  e.  S  ->  (
x R y  <->  ( z F x ) R ( z F y ) ) )
4 caovord.2 . . . . 5  |-  B  e. 
_V
5 caovord2.com . . . . 5  |-  ( x F y )  =  ( y F x )
61, 2, 3, 4, 5caovord2 6261 . . . 4  |-  ( B  e.  S  ->  ( A R C  <->  ( A F B ) R ( C F B ) ) )
76adantr 453 . . 3  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( A R C  <-> 
( A F B ) R ( C F B ) ) )
8 breq1 4217 . . 3  |-  ( ( A F B )  =  ( C F D )  ->  (
( A F B ) R ( C F B )  <->  ( C F D ) R ( C F B ) ) )
97, 8sylan9bb 682 . 2  |-  ( ( ( B  e.  S  /\  C  e.  S
)  /\  ( A F B )  =  ( C F D ) )  ->  ( A R C  <->  ( C F D ) R ( C F B ) ) )
10 caovord3.4 . . . 4  |-  D  e. 
_V
1110, 4, 3caovord 6260 . . 3  |-  ( C  e.  S  ->  ( D R B  <->  ( C F D ) R ( C F B ) ) )
1211ad2antlr 709 . 2  |-  ( ( ( B  e.  S  /\  C  e.  S
)  /\  ( A F B )  =  ( C F D ) )  ->  ( D R B  <->  ( C F D ) R ( C F B ) ) )
139, 12bitr4d 249 1  |-  ( ( ( B  e.  S  /\  C  e.  S
)  /\  ( A F B )  =  ( C F D ) )  ->  ( A R C  <->  D R B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   class class class wbr 4214  (class class class)co 6083
This theorem is referenced by:  genpnnp  8884  ltsrpr  8954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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