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Theorem swopolem 4323
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 2636 . 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 4026 . . . 4  |-  ( x  =  X  ->  (
x R y  <->  X R
y ) )
4 breq1 4026 . . . . 5  |-  ( x  =  X  ->  (
x R z  <->  X R
z ) )
54orbi1d 683 . . . 4  |-  ( x  =  X  ->  (
( x R z  \/  z R y )  <->  ( X R z  \/  z R y ) ) )
63, 5imbi12d 311 . . 3  |-  ( x  =  X  ->  (
( x R y  ->  ( x R z  \/  z R y ) )  <->  ( X R y  ->  ( X R z  \/  z R y ) ) ) )
7 breq2 4027 . . . 4  |-  ( y  =  Y  ->  ( X R y  <->  X R Y ) )
8 breq2 4027 . . . . 5  |-  ( y  =  Y  ->  (
z R y  <->  z R Y ) )
98orbi2d 682 . . . 4  |-  ( y  =  Y  ->  (
( X R z  \/  z R y )  <->  ( X R z  \/  z R Y ) ) )
107, 9imbi12d 311 . . 3  |-  ( y  =  Y  ->  (
( X R y  ->  ( X R z  \/  z R y ) )  <->  ( X R Y  ->  ( X R z  \/  z R Y ) ) ) )
11 breq2 4027 . . . . 5  |-  ( z  =  Z  ->  ( X R z  <->  X R Z ) )
12 breq1 4026 . . . . 5  |-  ( z  =  Z  ->  (
z R Y  <->  Z R Y ) )
1311, 12orbi12d 690 . . . 4  |-  ( z  =  Z  ->  (
( X R z  \/  z R Y )  <->  ( X R Z  \/  Z R Y ) ) )
1413imbi2d 307 . . 3  |-  ( z  =  Z  ->  (
( X R Y  ->  ( X R z  \/  z R Y ) )  <->  ( X R Y  ->  ( X R Z  \/  Z R Y ) ) ) )
156, 10, 14rspc3v 2893 . 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 455 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    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023
This theorem is referenced by:  swoer  6688  swoord1  6689  swoord2  6690
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-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-br 4024
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