Theorem caovassd 6214

Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovassg.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
caovassd.2	|- ( ph -> A e. S )
caovassd.3	|- ( ph -> B e. S )
caovassd.4	|- ( ph -> C e. S )

Assertion

Ref	Expression
caovassd	|- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )

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 caovassd

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( ph -> ph )
2		caovassd.2	. 2 |- ( ph -> A e. S )
3		caovassd.3	. 2 |- ( ph -> B e. S )
4		caovassd.4	. 2 |- ( ph -> C e. S )
5		caovassg.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
6	5	caovassg 6213	. 2 |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
7	1, 2, 3, 4, 6	syl13anc 1276	1 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203  (class class class)co 6050

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053

This theorem is referenced by:  caov32d  6235  caov12d  6236  caov13d  6238  caov4d  6239  caovdilemd  6246  caovimo  6248  enq0tr  7749  prarloclemlo  7809  ltsosr  8079  seqf1oglem2a  10880  grpinvalem  13598  grpinva  13599  grprida  13600  grprcan  13750