ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caovord3 Unicode version

Theorem caovord3 5818
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 5817 . . . 4  |-  ( B  e.  S  ->  ( A R C  <->  ( A F B ) R ( C F B ) ) )
76adantr 270 . . 3  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( A R C  <-> 
( A F B ) R ( C F B ) ) )
8 breq1 3848 . . 3  |-  ( ( A F B )  =  ( C F D )  ->  (
( A F B ) R ( C F B )  <->  ( C F D ) R ( C F B ) ) )
97, 8sylan9bb 450 . 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 5816 . . 3  |-  ( C  e.  S  ->  ( D R B  <->  ( C F D ) R ( C F B ) ) )
1211ad2antlr 473 . 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 189 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619   class class class wbr 3845  (class class class)co 5652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator