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Metamath Proof Explorer

Theorem swopolem

Description: Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014)

Ref Expression
Hypothesis swopolem.1 
|- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( x R y -> ( x R z \/ z R y ) ) )
Assertion swopolem 
|- ( ( ph /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( X R Y -> ( X R Z \/ Z R Y ) ) )

Proof

Step Hyp Ref Expression
1 swopolem.1 
 |-  ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( x R y -> ( x R z \/ z R y ) ) )
2 1 ralrimivvva 
 |-  ( ph -> A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) )
3 breq1 
 |-  ( x = X -> ( x R y <-> X R y ) )
4 breq1 
 |-  ( x = X -> ( x R z <-> X R z ) )
5 4 orbi1d 
 |-  ( x = X -> ( ( x R z \/ z R y ) <-> ( X R z \/ z R y ) ) )
6 3 5 imbi12d 
 |-  ( x = X -> ( ( x R y -> ( x R z \/ z R y ) ) <-> ( X R y -> ( X R z \/ z R y ) ) ) )
7 breq2 
 |-  ( y = Y -> ( X R y <-> X R Y ) )
8 breq2 
 |-  ( y = Y -> ( z R y <-> z R Y ) )
9 8 orbi2d 
 |-  ( y = Y -> ( ( X R z \/ z R y ) <-> ( X R z \/ z R Y ) ) )
10 7 9 imbi12d 
 |-  ( y = Y -> ( ( X R y -> ( X R z \/ z R y ) ) <-> ( X R Y -> ( X R z \/ z R Y ) ) ) )
11 breq2 
 |-  ( z = Z -> ( X R z <-> X R Z ) )
12 breq1 
 |-  ( z = Z -> ( z R Y <-> Z R Y ) )
13 11 12 orbi12d 
 |-  ( z = Z -> ( ( X R z \/ z R Y ) <-> ( X R Z \/ Z R Y ) ) )
14 13 imbi2d 
 |-  ( z = Z -> ( ( X R Y -> ( X R z \/ z R Y ) ) <-> ( X R Y -> ( X R Z \/ Z R Y ) ) ) )
15 6 10 14 rspc3v 
 |-  ( ( 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 ) ) ) )
16 2 15 mpan9 
 |-  ( ( ph /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( X R Y -> ( X R Z \/ Z R Y ) ) )