Theorem caov31 6066

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 6037	. . 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 6065	. . 3 |- ( A F ( C F B ) ) = ( C F ( A F B ) )
8	5, 7	eqtri 2198	. 2 |- ( ( A F C ) F B ) = ( C F ( A F B ) )
9	1, 3, 2, 6, 4	caov32 6064	. 2 |- ( ( A F B ) F C ) = ( ( A F C ) F B )
10	2, 1, 3, 6, 4	caov32 6064	. . 3 |- ( ( C F A ) F B ) = ( ( C F B ) F A )
11	2, 1, 3, 4	caovass 6037	. . 3 |- ( ( C F A ) F B ) = ( C F ( A F B ) )
12	10, 11	eqtr3i 2200	. 2 |- ( ( C F B ) F A ) = ( C F ( A F B ) )
13	8, 9, 12	3eqtr4i 2208	1 |- ( ( A F B ) F C ) = ( ( C F B ) F A )

Syntax hints:   = wceq 1353   e. wcel 2148   _Vcvv 2739  (class class class)co 5877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by:  caov13  6067