Theorem caov31 5710

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

Hypotheses

Ref	Expression
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

Ref	Expression
caov31	|- ( ( A F B ) F C ) = ( ( C F B ) F A )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, F, y, z

Proof of Theorem caov31

Step	Hyp	Ref	Expression
1		caov.1	. . . 4 |- A e. _V
2		caov.3	. . . 4 |- C e. _V
3		caov.2	. . . 4 |- B e. _V
4		caov.ass	. . . 4 |- ( ( x F y ) F z ) = ( x F ( y F z ) )
5	1, 2, 3, 4	caovass 5681	. . 3 |- ( ( A F C ) F B ) = ( A F ( C F B ) )
6		caov.com	. . . 4 |- ( x F y ) = ( y F x )
7	1, 2, 3, 6, 4	caov12 5709	. . 3 |- ( A F ( C F B ) ) = ( C F ( A F B ) )
8	5, 7	eqtri 2101	. 2 |- ( ( A F C ) F B ) = ( C F ( A F B ) )
9	1, 3, 2, 6, 4	caov32 5708	. 2 |- ( ( A F B ) F C ) = ( ( A F C ) F B )
10	2, 1, 3, 6, 4	caov32 5708	. . 3 |- ( ( C F A ) F B ) = ( ( C F B ) F A )
11	2, 1, 3, 4	caovass 5681	. . 3 |- ( ( C F A ) F B ) = ( C F ( A F B ) )
12	10, 11	eqtr3i 2103	. 2 |- ( ( C F B ) F A ) = ( C F ( A F B ) )
13	8, 9, 12	3eqtr4i 2111	1 |- ( ( A F B ) F C ) = ( ( C F B ) F A )

Syntax hints:   = wceq 1284   e. wcel 1433   _Vcvv 2601  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  caov13  5711