Theorem caov12d 6214

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 6189	. . 3 |- ( ph -> ( A F B ) = ( B F A ) )
5	4	oveq1d 6043	. 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 6192	. 2 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9	6, 3, 2, 7	caovassd 6192	. 2 |- ( ph -> ( ( B F A ) F C ) = ( B F ( A F C ) ) )
10	5, 8, 9	3eqtr3d 2272	1 |- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202  (class class class)co 6028

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  caov4d  6217  caovimo  6226  ltaddnq  7687  ltexnqq  7688  enq0tr  7714  mullocprlem  7850  1idprl  7870  1idpru  7871  cauappcvgprlemdisj  7931  mulcmpblnrlemg  8020  lttrsr  8042  ltsosr  8044  0idsr  8047  1idsr  8048  recexgt0sr  8053  mulgt0sr  8058  axmulass  8153