Theorem caov31 6031

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 6002	. . 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 6030	. . 3 |- ( A F ( C F B ) ) = ( C F ( A F B ) )
8	5, 7	eqtri 2186	. 2 |- ( ( A F C ) F B ) = ( C F ( A F B ) )
9	1, 3, 2, 6, 4	caov32 6029	. 2 |- ( ( A F B ) F C ) = ( ( A F C ) F B )
10	2, 1, 3, 6, 4	caov32 6029	. . 3 |- ( ( C F A ) F B ) = ( ( C F B ) F A )
11	2, 1, 3, 4	caovass 6002	. . 3 |- ( ( C F A ) F B ) = ( C F ( A F B ) )
12	10, 11	eqtr3i 2188	. 2 |- ( ( C F B ) F A ) = ( C F ( A F B ) )
13	8, 9, 12	3eqtr4i 2196	1 |- ( ( A F B ) F C ) = ( ( C F B ) F A )

Syntax hints:   = wceq 1343   e. wcel 2136   _Vcvv 2726  (class class class)co 5842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  caov13  6032