Theorem caov13d 6036

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
caov13d	|- ( 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 caov13d

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	1, 2, 3, 4, 5	caov31d 6035	. 2 |- ( ph -> ( ( A F B ) F C ) = ( ( C F B ) F A ) )
7	5, 1, 2, 3	caovassd 6012	. 2 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
8	5, 3, 2, 1	caovassd 6012	. 2 |- ( ph -> ( ( C F B ) F A ) = ( C F ( B F A ) ) )
9	6, 7, 8	3eqtr3d 2211	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:  enq0tr  7396  mullocprlem  7532  mulcmpblnrlemg  7702