Theorem caov12d 6130

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 6105	. . 3 |- ( ph -> ( A F B ) = ( B F A ) )
5	4	oveq1d 5961	. 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 6108	. 2 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9	6, 3, 2, 7	caovassd 6108	. 2 |- ( ph -> ( ( B F A ) F C ) = ( B F ( A F C ) ) )
10	5, 8, 9	3eqtr3d 2246	1 |- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2176  (class class class)co 5946

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-iota 5233  df-fv 5280  df-ov 5949

This theorem is referenced by:  caov4d  6133  caovimo  6142  ltaddnq  7522  ltexnqq  7523  enq0tr  7549  mullocprlem  7685  1idprl  7705  1idpru  7706  cauappcvgprlemdisj  7766  mulcmpblnrlemg  7855  lttrsr  7877  ltsosr  7879  0idsr  7882  1idsr  7883  recexgt0sr  7888  mulgt0sr  7893  axmulass  7988