Theorem caov411d 6275

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 ) ) )
caovd.4	|- ( ph -> D e. S )
caovd.cl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )

Assertion

Ref	Expression
caov411d	|- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, D, y, z   ph, x, y, z   x, F, y, z   x, S, y, z

Proof of Theorem caov411d

Step	Hyp	Ref	Expression
1		caovd.2	. . 3 |- ( ph -> B e. S )
2		caovd.1	. . 3 |- ( ph -> A e. S )
3		caovd.3	. . 3 |- ( ph -> C e. S )
4		caovd.com	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
5		caovd.ass	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
6		caovd.4	. . 3 |- ( ph -> D e. S )
7		caovd.cl	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
8	1, 2, 3, 4, 5, 6, 7	caov4d 6274	. 2 |- ( ph -> ( ( B F A ) F ( C F D ) ) = ( ( B F C ) F ( A F D ) ) )
9	4, 1, 2	caovcomd 6246	. . 3 |- ( ph -> ( B F A ) = ( A F B ) )
10	9	oveq1d 6100	. 2 |- ( ph -> ( ( B F A ) F ( C F D ) ) = ( ( A F B ) F ( C F D ) ) )
11	4, 1, 3	caovcomd 6246	. . 3 |- ( ph -> ( B F C ) = ( C F B ) )
12	11	oveq1d 6100	. 2 |- ( ph -> ( ( B F C ) F ( A F D ) ) = ( ( C F B ) F ( A F D ) ) )
13	8, 10, 12	3eqtr3d 2279	1 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by:  ecopovtrn  6906  ecopovtrng  6909  ltsonq  7765  ltanqg  7767  mulextsr1lem  8147