Theorem caov32 5958

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
caov32	|- ( ( A F B ) F C ) = ( ( A F C ) F B )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, F, y, z

Proof of Theorem caov32

Step	Hyp	Ref	Expression
1		caov.2	. . . 4 |- B e. _V
2		caov.3	. . . 4 |- C e. _V
3		caov.com	. . . 4 |- ( x F y ) = ( y F x )
4	1, 2, 3	caovcom 5928	. . 3 |- ( B F C ) = ( C F B )
5	4	oveq2i 5785	. 2 |- ( A F ( B F C ) ) = ( A F ( C F B ) )
6		caov.1	. . 3 |- A e. _V
7		caov.ass	. . 3 |- ( ( x F y ) F z ) = ( x F ( y F z ) )
8	6, 1, 2, 7	caovass 5931	. 2 |- ( ( A F B ) F C ) = ( A F ( B F C ) )
9	6, 2, 1, 7	caovass 5931	. 2 |- ( ( A F C ) F B ) = ( A F ( C F B ) )
10	5, 8, 9	3eqtr4i 2170	1 |- ( ( A F B ) F C ) = ( ( A F C ) F B )

Syntax hints:   = wceq 1331   e. wcel 1480   _Vcvv 2686  (class class class)co 5774

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777

This theorem is referenced by:  caov31  5960