Theorem caov42d 6039

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
caov42d	|- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( D F B ) ) )

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 caov42d

Step	Hyp	Ref	Expression
1		caovd.1	. . 3 |- ( ph -> A e. S )
2		caovd.2	. . 3 |- ( ph -> B 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 6037	. 2 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )
9	4, 2, 6	caovcomd 6009	. . 3 |- ( ph -> ( B F D ) = ( D F B ) )
10	9	oveq2d 5869	. 2 |- ( ph -> ( ( A F C ) F ( B F D ) ) = ( ( A F C ) F ( D F B ) ) )
11	8, 10	eqtrd 2203	1 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( D F B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141  (class class class)co 5853

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  caovlem2d  6045  mulcmpblnrlemg  7702  ltasrg  7732  axmulass  7835