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Theorem caovord3d 6030
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 4026 . 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 6029 . . 3  |-  ( ph  ->  ( A R C  <-> 
( A F B ) R ( C F B ) ) )
8 caovord3d.5 . . . 4  |-  ( ph  ->  D  e.  S )
92, 8, 5, 4caovordd 6028 . . 3  |-  ( ph  ->  ( D R B  <-> 
( C F D ) R ( C F B ) ) )
107, 9bibi12d 312 . 2  |-  ( ph  ->  ( ( A R C  <->  D R B )  <-> 
( ( A F B ) R ( C F B )  <-> 
( C F D ) R ( C F B ) ) ) )
111, 10syl5ibr 212 1  |-  ( ph  ->  ( ( 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    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858
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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