Theorem caov42d 6192

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovd.1	|- ( ph -> A e. S )
caovd.2	|- ( ph -> B e. S )
caovd.3	|- ( ph -> C e. S )
caovd.com	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
caovd.ass	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
caovd.4	|- ( ph -> D e. S )
caovd.cl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )

Assertion

Ref	Expression
caov42d	|- ( ph -> ( ( 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   ph, x, y, z   x, F, y, z   x, S, y, z

Proof of Theorem caov42d

Step	Hyp	Ref	Expression
1		caovd.1	. . 3 |- ( ph -> A e. S )
2		caovd.2	. . 3 |- ( ph -> B e. S )
3		caovd.3	. . 3 |- ( ph -> C e. S )
4		caovd.com	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
5		caovd.ass	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
6		caovd.4	. . 3 |- ( ph -> D e. S )
7		caovd.cl	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
8	1, 2, 3, 4, 5, 6, 7	caov4d 6190	. 2 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )
9	4, 2, 6	caovcomd 6162	. . 3 |- ( ph -> ( B F D ) = ( D F B ) )
10	9	oveq2d 6017	. 2 |- ( ph -> ( ( A F C ) F ( B F D ) ) = ( ( A F C ) F ( D F B ) ) )
11	8, 10	eqtrd 2262	1 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( D F B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200  (class class class)co 6001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004

This theorem is referenced by:  caovlem2d  6198  mulcmpblnrlemg  7927  ltasrg  7957  axmulass  8060