Theorem caov31 6042

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 6013	. . 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 6041	. . 3 |- ( A F ( C F B ) ) = ( C F ( A F B ) )
8	5, 7	eqtri 2191	. 2 |- ( ( A F C ) F B ) = ( C F ( A F B ) )
9	1, 3, 2, 6, 4	caov32 6040	. 2 |- ( ( A F B ) F C ) = ( ( A F C ) F B )
10	2, 1, 3, 6, 4	caov32 6040	. . 3 |- ( ( C F A ) F B ) = ( ( C F B ) F A )
11	2, 1, 3, 4	caovass 6013	. . 3 |- ( ( C F A ) F B ) = ( C F ( A F B ) )
12	10, 11	eqtr3i 2193	. 2 |- ( ( C F B ) F A ) = ( C F ( A F B ) )
13	8, 9, 12	3eqtr4i 2201	1 |- ( ( A F B ) F C ) = ( ( C F B ) F A )

Syntax hints:   = wceq 1348   e. wcel 2141   _Vcvv 2730  (class class class)co 5853

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  caov13  6043