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Theorem caovord3d 6470
Description: Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovordg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  <-> 
( z F x ) R ( z F y ) ) )
caovordd.2  |-  ( ph  ->  A  e.  S )
caovordd.3  |-  ( ph  ->  B  e.  S )
caovordd.4  |-  ( ph  ->  C  e.  S )
caovord2d.com  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
caovord3d.5  |-  ( ph  ->  D  e.  S )
Assertion
Ref Expression
caovord3d  |-  ( ph  ->  ( ( 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    ph, x, y, z   
x, F, y, z   
x, R, y, z   
x, S, y, z

Proof of Theorem caovord3d
StepHypRef Expression
1 breq1 4440 . 2  |-  ( ( A F B )  =  ( C F D )  ->  (
( A F B ) R ( C F B )  <->  ( C F D ) R ( C F B ) ) )
2 caovordg.1 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  <-> 
( z F x ) R ( z F y ) ) )
3 caovordd.2 . . . 4  |-  ( ph  ->  A  e.  S )
4 caovordd.4 . . . 4  |-  ( ph  ->  C  e.  S )
5 caovordd.3 . . . 4  |-  ( ph  ->  B  e.  S )
6 caovord2d.com . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
72, 3, 4, 5, 6caovord2d 6469 . . 3  |-  ( ph  ->  ( A R C  <-> 
( A F B ) R ( C F B ) ) )
8 caovord3d.5 . . . 4  |-  ( ph  ->  D  e.  S )
92, 8, 5, 4caovordd 6468 . . 3  |-  ( ph  ->  ( D R B  <-> 
( C F D ) R ( C F B ) ) )
107, 9bibi12d 321 . 2  |-  ( ph  ->  ( ( A R C  <->  D R B )  <-> 
( ( A F B ) R ( C F B )  <-> 
( C F D ) R ( C F B ) ) ) )
111, 10syl5ibr 221 1  |-  ( ph  ->  ( ( 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    /\ w3a 974    = wceq 1383    e. wcel 1804   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: (None)
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