Theorem caov31d 6035

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
caov31d	|- ( ph -> ( ( A F B ) F C ) = ( ( C F B ) F A ) )

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 caov31d

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.3	. . . 4 |- ( ph -> C e. S )
4	1, 2, 3	caovcomd 6009	. . 3 |- ( ph -> ( A F C ) = ( C F A ) )
5	4	oveq1d 5868	. 2 |- ( ph -> ( ( A F C ) F B ) = ( ( C F A ) F B ) )
6		caovd.2	. . 3 |- ( ph -> B e. S )
7		caovd.ass	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
8	2, 6, 3, 1, 7	caov32d 6033	. 2 |- ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) )
9	3, 6, 2, 1, 7	caov32d 6033	. 2 |- ( ph -> ( ( C F B ) F A ) = ( ( C F A ) F B ) )
10	5, 8, 9	3eqtr4d 2213	1 |- ( ph -> ( ( A F B ) F C ) = ( ( C F B ) F A ) )

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:  caov13d  6036