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Metamath Proof Explorer

Theorem caov31

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995)

Ref Expression
Hypotheses caov.1 
|- A e. _V
caov.2 
|- B e. _V
caov.3 
|- C e. _V
caov.com 
|- ( x F y ) = ( y F x )
caov.ass 
|- ( ( x F y ) F z ) = ( x F ( y F z ) )
Assertion caov31 
|- ( ( A F B ) F C ) = ( ( C F B ) F A )

Proof

Step Hyp Ref Expression
1 caov.1 
 |-  A e. _V
2 caov.2 
 |-  B e. _V
3 caov.3 
 |-  C e. _V
4 caov.com 
 |-  ( x F y ) = ( y F x )
5 caov.ass 
 |-  ( ( x F y ) F z ) = ( x F ( y F z ) )
6 1 3 2 5 caovass 
 |-  ( ( A F C ) F B ) = ( A F ( C F B ) )
7 1 3 2 4 5 caov12 
 |-  ( A F ( C F B ) ) = ( C F ( A F B ) )
8 6 7 eqtri 
 |-  ( ( A F C ) F B ) = ( C F ( A F B ) )
9 1 2 3 4 5 caov32 
 |-  ( ( A F B ) F C ) = ( ( A F C ) F B )
10 3 1 2 4 5 caov32 
 |-  ( ( C F A ) F B ) = ( ( C F B ) F A )
11 3 1 2 5 caovass 
 |-  ( ( C F A ) F B ) = ( C F ( A F B ) )
12 10 11 eqtr3i 
 |-  ( ( C F B ) F A ) = ( C F ( A F B ) )
13 8 9 12 3eqtr4i 
 |-  ( ( A F B ) F C ) = ( ( C F B ) F A )