Theorem caov42d 6051

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 6049	. 2 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )
9	4, 2, 6	caovcomd 6021	. . 3 |- ( ph -> ( B F D ) = ( D F B ) )
10	9	oveq2d 5881	. 2 |- ( ph -> ( ( A F C ) F ( B F D ) ) = ( ( A F C ) F ( D F B ) ) )
11	8, 10	eqtrd 2208	1 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( D F B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2146  (class class class)co 5865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868

This theorem is referenced by:  caovlem2d  6057  mulcmpblnrlemg  7714  ltasrg  7744  axmulass  7847