Theorem caov32d 5959

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 5935	. . 3 |- ( ph -> ( B F C ) = ( C F B ) )
5	4	oveq2d 5798	. 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 5938	. 2 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9	6, 7, 3, 2	caovassd 5938	. 2 |- ( ph -> ( ( A F C ) F B ) = ( A F ( C F B ) ) )
10	5, 8, 9	3eqtr4d 2183	1 |- ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1332   e. wcel 1481  (class class class)co 5782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by:  caov31d  5961  mulcanenq  7217  mulcanenq0ec  7277  ltexprlemrl  7442  ltexprlemru  7444  cauappcvgprlemladdfl  7487  cauappcvgprlemladdru  7488  mulcmpblnrlemg  7572  ltsosr  7596  recexgt0sr  7605  mulgt0sr  7610  caucvgsrlemoffcau  7630  caucvgsrlemoffres  7632  resqrexlemover  10814