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Metamath Proof Explorer

Theorem caovord2d

Description: Operation ordering law with commuted arguments. (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 ) )
Assertion caovord2d 
|- ( ph -> ( A R B <-> ( A F C ) R ( B F C ) ) )

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 1 2 3 4 caovordd 
 |-  ( ph -> ( A R B <-> ( C F A ) R ( C F B ) ) )
7 5 4 2 caovcomd 
 |-  ( ph -> ( C F A ) = ( A F C ) )
8 5 4 3 caovcomd 
 |-  ( ph -> ( C F B ) = ( B F C ) )
9 7 8 breq12d 
 |-  ( ph -> ( ( C F A ) R ( C F B ) <-> ( A F C ) R ( B F C ) ) )
10 6 9 bitrd 
 |-  ( ph -> ( A R B <-> ( A F C ) R ( B F C ) ) )