Theorem caov31 6138

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 6109	. . 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 6137	. . 3 |- ( A F ( C F B ) ) = ( C F ( A F B ) )
8	5, 7	eqtri 2226	. 2 |- ( ( A F C ) F B ) = ( C F ( A F B ) )
9	1, 3, 2, 6, 4	caov32 6136	. 2 |- ( ( A F B ) F C ) = ( ( A F C ) F B )
10	2, 1, 3, 6, 4	caov32 6136	. . 3 |- ( ( C F A ) F B ) = ( ( C F B ) F A )
11	2, 1, 3, 4	caovass 6109	. . 3 |- ( ( C F A ) F B ) = ( C F ( A F B ) )
12	10, 11	eqtr3i 2228	. 2 |- ( ( C F B ) F A ) = ( C F ( A F B ) )
13	8, 9, 12	3eqtr4i 2236	1 |- ( ( A F B ) F C ) = ( ( C F B ) F A )

Syntax hints:   = wceq 1373   e. wcel 2176   _Vcvv 2772  (class class class)co 5946

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-iota 5233  df-fv 5280  df-ov 5949

This theorem is referenced by:  caov13  6139