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Theorem caovord3 6033
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 6032 . . . 4  |-  ( B  e.  S  ->  ( A R C  <->  ( A F B ) R ( C F B ) ) )
76adantr 451 . . 3  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( A R C  <-> 
( A F B ) R ( C F B ) ) )
8 breq1 4026 . . 3  |-  ( ( A F B )  =  ( C F D )  ->  (
( A F B ) R ( C F B )  <->  ( C F D ) R ( C F B ) ) )
97, 8sylan9bb 680 . 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 6031 . . 3  |-  ( C  e.  S  ->  ( D R B  <->  ( C F D ) R ( C F B ) ) )
1211ad2antlr 707 . 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 247 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023  (class class class)co 5858
This theorem is referenced by:  genpnnp  8629  ltsrpr  8699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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