Theorem caov411d 6027

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 6026	. 2 |- ( ph -> ( ( B F A ) F ( C F D ) ) = ( ( B F C ) F ( A F D ) ) )
9	4, 1, 2	caovcomd 5998	. . 3 |- ( ph -> ( B F A ) = ( A F B ) )
10	9	oveq1d 5857	. 2 |- ( ph -> ( ( B F A ) F ( C F D ) ) = ( ( A F B ) F ( C F D ) ) )
11	4, 1, 3	caovcomd 5998	. . 3 |- ( ph -> ( B F C ) = ( C F B ) )
12	11	oveq1d 5857	. 2 |- ( ph -> ( ( B F C ) F ( A F D ) ) = ( ( C F B ) F ( A F D ) ) )
13	8, 10, 12	3eqtr3d 2206	1 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136  (class class class)co 5842

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  ecopovtrn  6598  ecopovtrng  6601  ltsonq  7339  ltanqg  7341  mulextsr1lem  7721