Theorem caov12d 5960

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 5935	. . 3 |- ( ph -> ( A F B ) = ( B F A ) )
5	4	oveq1d 5797	. 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 5938	. 2 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9	6, 3, 2, 7	caovassd 5938	. 2 |- ( ph -> ( ( B F A ) F C ) = ( B F ( A F C ) ) )
10	5, 8, 9	3eqtr3d 2181	1 |- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) )

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:  caov4d  5963  caovimo  5972  ltaddnq  7239  ltexnqq  7240  enq0tr  7266  mullocprlem  7402  1idprl  7422  1idpru  7423  cauappcvgprlemdisj  7483  mulcmpblnrlemg  7572  lttrsr  7594  ltsosr  7596  0idsr  7599  1idsr  7600  recexgt0sr  7605  mulgt0sr  7610  axmulass  7705