Theorem caov12d 6105

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 6080	. . 3 |- ( ph -> ( A F B ) = ( B F A ) )
5	4	oveq1d 5937	. 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 6083	. 2 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9	6, 3, 2, 7	caovassd 6083	. 2 |- ( ph -> ( ( B F A ) F C ) = ( B F ( A F C ) ) )
10	5, 8, 9	3eqtr3d 2237	1 |- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167  (class class class)co 5922

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  caov4d  6108  caovimo  6117  ltaddnq  7474  ltexnqq  7475  enq0tr  7501  mullocprlem  7637  1idprl  7657  1idpru  7658  cauappcvgprlemdisj  7718  mulcmpblnrlemg  7807  lttrsr  7829  ltsosr  7831  0idsr  7834  1idsr  7835  recexgt0sr  7840  mulgt0sr  7845  axmulass  7940