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Theorem caovord3 5912
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 5911 . . . 4  |-  ( B  e.  S  ->  ( A R C  <->  ( A F B ) R ( C F B ) ) )
76adantr 274 . . 3  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( A R C  <-> 
( A F B ) R ( C F B ) ) )
8 breq1 3902 . . 3  |-  ( ( A F B )  =  ( C F D )  ->  (
( A F B ) R ( C F B )  <->  ( C F D ) R ( C F B ) ) )
97, 8sylan9bb 457 . 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 5910 . . 3  |-  ( C  e.  S  ->  ( D R B  <->  ( C F D ) R ( C F B ) ) )
1211ad2antlr 480 . 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 190 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    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   _Vcvv 2660   class class class wbr 3899  (class class class)co 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by: (None)
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