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Metamath Proof Explorer

Theorem caovcomg

Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013)

Ref Expression
Hypothesis caovcomg.1 
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
Assertion caovcomg 
|- ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )

Proof

Step Hyp Ref Expression
1 caovcomg.1 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
2 1 ralrimivva 
 |-  ( ph -> A. x e. S A. y e. S ( x F y ) = ( y F x ) )
3 oveq1 
 |-  ( x = A -> ( x F y ) = ( A F y ) )
4 oveq2 
 |-  ( x = A -> ( y F x ) = ( y F A ) )
5 3 4 eqeq12d 
 |-  ( x = A -> ( ( x F y ) = ( y F x ) <-> ( A F y ) = ( y F A ) ) )
6 oveq2 
 |-  ( y = B -> ( A F y ) = ( A F B ) )
7 oveq1 
 |-  ( y = B -> ( y F A ) = ( B F A ) )
8 6 7 eqeq12d 
 |-  ( y = B -> ( ( A F y ) = ( y F A ) <-> ( A F B ) = ( B F A ) ) )
9 5 8 rspc2v 
 |-  ( ( A e. S /\ B e. S ) -> ( A. x e. S A. y e. S ( x F y ) = ( y F x ) -> ( A F B ) = ( B F A ) ) )
10 2 9 mpan9 
 |-  ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )