Theorem caov42 6867

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 ) )
caov.4	|- D e. _V

Assertion

Ref	Expression
caov42	|- ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( D F B ) )

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

Proof of Theorem caov42

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		caov.4	. . 3 |- D e. _V
7	1, 2, 3, 4, 5, 6	caov4 6865	. 2 |- ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) )
8	2, 6, 4	caovcom 6831	. . 3 |- ( B F D ) = ( D F B )
9	8	oveq2i 6661	. 2 |- ( ( A F C ) F ( B F D ) ) = ( ( A F C ) F ( D F B ) )
10	7, 9	eqtri 2644	1 |- ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( D F B ) )

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  caovlem2  6870  mulcmpblnrlem  9891  ltasr  9921  axmulass  9978