Theorem caovassd 6106

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 6105	. 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 1252	1 |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2176  (class class class)co 5944

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  caov32d  6127  caov12d  6128  caov13d  6130  caov4d  6131  caovdilemd  6138  caovimo  6140  enq0tr  7547  prarloclemlo  7607  ltsosr  7877  seqf1oglem2a  10663  grpinvalem  13217  grpinva  13218  grprida  13219  grprcan  13369