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Theorem caovord3d 5941
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 3932 . 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 5940 . . 3  |-  ( ph  ->  ( A R C  <-> 
( A F B ) R ( C F B ) ) )
8 caovord3d.5 . . . 4  |-  ( ph  ->  D  e.  S )
92, 8, 5, 4caovordd 5939 . . 3  |-  ( ph  ->  ( D R B  <-> 
( C F D ) R ( C F B ) ) )
107, 9bibi12d 234 . 2  |-  ( ph  ->  ( ( A R C  <->  D R B )  <-> 
( ( A F B ) R ( C F B )  <-> 
( C F D ) R ( C F B ) ) ) )
111, 10syl5ibr 155 1  |-  ( ph  ->  ( ( 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    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  ordpipqqs  7189  ltsrprg  7562
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