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 ) ) ) |
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caovordd.2 | |- ( ph -> A e. S ) |
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caovordd.3 | |- ( ph -> B e. S ) |
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caovordd.4 | |- ( ph -> C e. S ) |
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caovord2d.com | |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) ) |
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Assertion | caovord2d | |- ( ph -> ( A R B <-> ( A F C ) R ( B F C ) ) ) |
Step | Hyp | Ref | Expression |
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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 ) |
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3 | caovordd.3 | |- ( ph -> B e. S ) |
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4 | caovordd.4 | |- ( ph -> C e. S ) |
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5 | caovord2d.com | |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) ) |
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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 ) ) ) |