Theorem caov32d 6243

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
caov32d	|- ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) )

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 caov32d

Step	Hyp	Ref	Expression
1		caovd.com	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
2		caovd.2	. . . 4 |- ( ph -> B e. S )
3		caovd.3	. . . 4 |- ( ph -> C e. S )
4	1, 2, 3	caovcomd 6219	. . 3 |- ( ph -> ( B F C ) = ( C F B ) )
5	4	oveq2d 6074	. 2 |- ( ph -> ( A F ( B F C ) ) = ( A F ( C F B ) ) )
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.1	. . 3 |- ( ph -> A e. S )
8	6, 7, 2, 3	caovassd 6222	. 2 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9	6, 7, 3, 2	caovassd 6222	. 2 |- ( ph -> ( ( A F C ) F B ) = ( A F ( C F B ) ) )
10	5, 8, 9	3eqtr4d 2277	1 |- ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205  (class class class)co 6058

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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061

This theorem is referenced by:  caov31d  6245  mulcanenq  7716  mulcanenq0ec  7776  ltexprlemrl  7941  ltexprlemru  7943  cauappcvgprlemladdfl  7986  cauappcvgprlemladdru  7987  mulcmpblnrlemg  8071  ltsosr  8095  recexgt0sr  8104  mulgt0sr  8109  caucvgsrlemoffcau  8129  caucvgsrlemoffres  8131  resqrexlemover  11720