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Metamath Proof Explorer

Theorem caovord3d

Description: Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014)

Ref Expression
Hypotheses 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 caovord3d 
|- ( ph -> ( ( A F B ) = ( C F D ) -> ( A R C <-> D R B ) ) )

Proof

Step Hyp Ref Expression
1 caovordg.1 
 |-  ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )
2 caovordd.2 
 |-  ( ph -> A e. S )
3 caovordd.3 
 |-  ( ph -> B e. S )
4 caovordd.4 
 |-  ( ph -> C e. S )
5 caovord2d.com 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
6 caovord3d.5 
 |-  ( ph -> D e. S )
7 breq1 
 |-  ( ( A F B ) = ( C F D ) -> ( ( A F B ) R ( C F B ) <-> ( C F D ) R ( C F B ) ) )
8 1 2 4 3 5 caovord2d 
 |-  ( ph -> ( A R C <-> ( A F B ) R ( C F B ) ) )
9 1 6 3 4 caovordd 
 |-  ( ph -> ( D R B <-> ( C F D ) R ( C F B ) ) )
10 8 9 bibi12d 
 |-  ( ph -> ( ( A R C <-> D R B ) <-> ( ( A F B ) R ( C F B ) <-> ( C F D ) R ( C F B ) ) ) )
11 7 10 imbitrrid 
 |-  ( ph -> ( ( A F B ) = ( C F D ) -> ( A R C <-> D R B ) ) )