Theorem caovassd 5660

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 5659	. 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 1137	1 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  caov32d  5681  caov12d  5682  caov13d  5684  caov4d  5685  caovdilemd  5692  caovimo  5694  grprinvlem  5695  grprinvd  5696  grpridd  5697  enq0tr  6532  prarloclemlo  6592  ltsosr  6849