Theorem caov13 5969

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
caov13	|- ( 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 caov13

Step	Hyp	Ref	Expression
1		caov.1	. . 3 |- A e. _V
2		caov.2	. . 3 |- B e. _V
3		caov.3	. . 3 |- C e. _V
4		caov.com	. . 3 |- ( x F y ) = ( y F x )
5		caov.ass	. . 3 |- ( ( x F y ) F z ) = ( x F ( y F z ) )
6	1, 2, 3, 4, 5	caov31 5968	. 2 |- ( ( A F B ) F C ) = ( ( C F B ) F A )
7	1, 2, 3, 5	caovass 5939	. 2 |- ( ( A F B ) F C ) = ( A F ( B F C ) )
8	3, 2, 1, 5	caovass 5939	. 2 |- ( ( C F B ) F A ) = ( C F ( B F A ) )
9	6, 7, 8	3eqtr3i 2169	1 |- ( A F ( B F C ) ) = ( C F ( B F A ) )

Syntax hints:   = wceq 1332   e. wcel 1481   _Vcvv 2689  (class class class)co 5782

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by: (None)