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Metamath Proof Explorer

Theorem caov411

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 ) )
caov.4 
|- D e. _V
Assertion caov411 
|- ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) )

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 caov.4 
 |-  D e. _V
7 1 2 3 4 5 caov31 
 |-  ( ( A F B ) F C ) = ( ( C F B ) F A )
8 7 oveq1i 
 |-  ( ( ( A F B ) F C ) F D ) = ( ( ( C F B ) F A ) F D )
9 ovex 
 |-  ( A F B ) e. _V
10 9 3 6 5 caovass 
 |-  ( ( ( A F B ) F C ) F D ) = ( ( A F B ) F ( C F D ) )
11 ovex 
 |-  ( C F B ) e. _V
12 11 1 6 5 caovass 
 |-  ( ( ( C F B ) F A ) F D ) = ( ( C F B ) F ( A F D ) )
13 8 10 12 3eqtr3i 
 |-  ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) )