Theorem caov12 5714

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
caov12	|- ( A F ( B F C ) ) = ( B F ( A F C ) )

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

Proof of Theorem caov12

Step	Hyp	Ref	Expression
1		caov.1	. . . 4 |- A e. _V
2		caov.2	. . . 4 |- B e. _V
3		caov.com	. . . 4 |- ( x F y ) = ( y F x )
4	1, 2, 3	caovcom 5683	. . 3 |- ( A F B ) = ( B F A )
5	4	oveq1i 5547	. 2 |- ( ( A F B ) F C ) = ( ( B F A ) F C )
6		caov.3	. . 3 |- C e. _V
7		caov.ass	. . 3 |- ( ( x F y ) F z ) = ( x F ( y F z ) )
8	1, 2, 6, 7	caovass 5686	. 2 |- ( ( A F B ) F C ) = ( A F ( B F C ) )
9	2, 1, 6, 7	caovass 5686	. 2 |- ( ( B F A ) F C ) = ( B F ( A F C ) )
10	5, 8, 9	3eqtr3i 2110	1 |- ( A F ( B F C ) ) = ( B F ( A F C ) )

Syntax hints:   = wceq 1285   e. wcel 1434   _Vcvv 2602  (class class class)co 5537

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-iota 4891  df-fv 4934  df-ov 5540

This theorem is referenced by:  caov31  5715