Theorem caov12 5689

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 5658	. . 3 |- ( A F B ) = ( B F A )
5	4	oveq1i 5522	. 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 5661	. 2 |- ( ( A F B ) F C ) = ( A F ( B F C ) )
9	2, 1, 6, 7	caovass 5661	. 2 |- ( ( B F A ) F C ) = ( B F ( A F C ) )
10	5, 8, 9	3eqtr3i 2068	1 |- ( A F ( B F C ) ) = ( B F ( A F C ) )

Syntax hints:   = wceq 1243   e. wcel 1393   _Vcvv 2557  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  caov31  5690