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Theorem caovord3 6473
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 6472 . . . 4  |-  ( B  e.  S  ->  ( A R C  <->  ( A F B ) R ( C F B ) ) )
76adantr 465 . . 3  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( A R C  <-> 
( A F B ) R ( C F B ) ) )
8 breq1 4440 . . 3  |-  ( ( A F B )  =  ( C F D )  ->  (
( A F B ) R ( C F B )  <->  ( C F D ) R ( C F B ) ) )
97, 8sylan9bb 699 . 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 6471 . . 3  |-  ( C  e.  S  ->  ( D R B  <->  ( C F D ) R ( C F B ) ) )
1211ad2antlr 726 . 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 256 1  |-  ( ( ( B  e.  S  /\  C  e.  S
)  /\  ( A F B )  =  ( C F D ) )  ->  ( A R C  <->  D R B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095   class class class wbr 4437  (class class class)co 6281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284
This theorem is referenced by:  genpnnp  9386  ltsrpr  9457
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