Theorem caov32d 5825

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 5801	. . 3 |- ( ph -> ( B F C ) = ( C F B ) )
5	4	oveq2d 5668	. 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 5804	. 2 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9	6, 7, 3, 2	caovassd 5804	. 2 |- ( ph -> ( ( A F C ) F B ) = ( A F ( C F B ) ) )
10	5, 8, 9	3eqtr4d 2130	1 |- ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 924   = wceq 1289   e. wcel 1438  (class class class)co 5652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655

This theorem is referenced by:  caov31d  5827  mulcanenq  6942  mulcanenq0ec  7002  ltexprlemrl  7167  ltexprlemru  7169  cauappcvgprlemladdfl  7212  cauappcvgprlemladdru  7213  mulcmpblnrlemg  7284  ltsosr  7308  recexgt0sr  7317  mulgt0sr  7321  caucvgsrlemoffcau  7341  caucvgsrlemoffres  7343  resqrexlemover  10439