Theorem caov32d 6077

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 6053	. . 3 |- ( ph -> ( B F C ) = ( C F B ) )
5	4	oveq2d 5912	. 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 6056	. 2 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9	6, 7, 3, 2	caovassd 6056	. 2 |- ( ph -> ( ( A F C ) F B ) = ( A F ( C F B ) ) )
10	5, 8, 9	3eqtr4d 2232	1 |- ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160  (class class class)co 5896

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5899

This theorem is referenced by:  caov31d  6079  mulcanenq  7414  mulcanenq0ec  7474  ltexprlemrl  7639  ltexprlemru  7641  cauappcvgprlemladdfl  7684  cauappcvgprlemladdru  7685  mulcmpblnrlemg  7769  ltsosr  7793  recexgt0sr  7802  mulgt0sr  7807  caucvgsrlemoffcau  7827  caucvgsrlemoffres  7829  resqrexlemover  11051