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Theorem swopolem 4089
Description: Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)
Hypothesis
Ref Expression
swopolem.1  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( x R y  ->  ( x R z  \/  z R y ) ) )
Assertion
Ref Expression
swopolem  |-  ( (
ph  /\  ( X  e.  A  /\  Y  e.  A  /\  Z  e.  A ) )  -> 
( X R Y  ->  ( X R Z  \/  Z R Y ) ) )
Distinct variable groups:    x, y, z, A    ph, x, y, z   
x, R, y, z   
x, X, y, z   
y, Y, z    z, Z
Allowed substitution hints:    Y( x)    Z( x, y)

Proof of Theorem swopolem
StepHypRef Expression
1 swopolem.1 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( x R y  ->  ( x R z  \/  z R y ) ) )
21ralrimivvva 2450 . 2  |-  ( ph  ->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x R y  ->  ( x R z  \/  z R y ) ) )
3 breq1 3809 . . . 4  |-  ( x  =  X  ->  (
x R y  <->  X R
y ) )
4 breq1 3809 . . . . 5  |-  ( x  =  X  ->  (
x R z  <->  X R
z ) )
54orbi1d 738 . . . 4  |-  ( x  =  X  ->  (
( x R z  \/  z R y )  <->  ( X R z  \/  z R y ) ) )
63, 5imbi12d 232 . . 3  |-  ( x  =  X  ->  (
( x R y  ->  ( x R z  \/  z R y ) )  <->  ( X R y  ->  ( X R z  \/  z R y ) ) ) )
7 breq2 3810 . . . 4  |-  ( y  =  Y  ->  ( X R y  <->  X R Y ) )
8 breq2 3810 . . . . 5  |-  ( y  =  Y  ->  (
z R y  <->  z R Y ) )
98orbi2d 737 . . . 4  |-  ( y  =  Y  ->  (
( X R z  \/  z R y )  <->  ( X R z  \/  z R Y ) ) )
107, 9imbi12d 232 . . 3  |-  ( y  =  Y  ->  (
( X R y  ->  ( X R z  \/  z R y ) )  <->  ( X R Y  ->  ( X R z  \/  z R Y ) ) ) )
11 breq2 3810 . . . . 5  |-  ( z  =  Z  ->  ( X R z  <->  X R Z ) )
12 breq1 3809 . . . . 5  |-  ( z  =  Z  ->  (
z R Y  <->  Z R Y ) )
1311, 12orbi12d 740 . . . 4  |-  ( z  =  Z  ->  (
( X R z  \/  z R Y )  <->  ( X R Z  \/  Z R Y ) ) )
1413imbi2d 228 . . 3  |-  ( z  =  Z  ->  (
( X R Y  ->  ( X R z  \/  z R Y ) )  <->  ( X R Y  ->  ( X R Z  \/  Z R Y ) ) ) )
156, 10, 14rspc3v 2725 . 2  |-  ( ( X  e.  A  /\  Y  e.  A  /\  Z  e.  A )  ->  ( A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x R y  ->  ( x R z  \/  z R y ) )  ->  ( X R Y  ->  ( X R Z  \/  Z R Y ) ) ) )
162, 15mpan9 275 1  |-  ( (
ph  /\  ( X  e.  A  /\  Y  e.  A  /\  Z  e.  A ) )  -> 
( X R Y  ->  ( X R Z  \/  Z R Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2353   class class class wbr 3806
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2613  df-un 2987  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807
This theorem is referenced by:  swoer  6223  swoord1  6224  swoord2  6225
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