Theorem caov12d 5702

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 ) ) )

Assertion

Ref	Expression
caov12d	|- ( ph -> ( 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   ph, x, y, z   x, F, y, z   x, S, y, z

Proof of Theorem caov12d

Step	Hyp	Ref	Expression
1		caovd.com	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
2		caovd.1	. . . 4 |- ( ph -> A e. S )
3		caovd.2	. . . 4 |- ( ph -> B e. S )
4	1, 2, 3	caovcomd 5677	. . 3 |- ( ph -> ( A F B ) = ( B F A ) )
5	4	oveq1d 5547	. 2 |- ( ph -> ( ( A F B ) F C ) = ( ( B F A ) F C ) )
6		caovd.ass	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
7		caovd.3	. . 3 |- ( ph -> C e. S )
8	6, 2, 3, 7	caovassd 5680	. 2 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9	6, 3, 2, 7	caovassd 5680	. 2 |- ( ph -> ( ( B F A ) F C ) = ( B F ( A F C ) ) )
10	5, 8, 9	3eqtr3d 2121	1 |- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433  (class class class)co 5532

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  caov4d  5705  caovimo  5714  ltaddnq  6597  ltexnqq  6598  enq0tr  6624  mullocprlem  6760  1idprl  6780  1idpru  6781  cauappcvgprlemdisj  6841  mulcmpblnrlemg  6917  lttrsr  6939  ltsosr  6941  0idsr  6944  1idsr  6945  recexgt0sr  6950  mulgt0sr  6954  axmulass  7039