Theorem caovass 6166

Description: Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)

Hypotheses

Ref	Expression
caovass.1	|- A e. _V
caovass.2	|- B e. _V
caovass.3	|- C e. _V
caovass.4	|- ( ( x F y ) F z ) = ( x F ( y F z ) )

Assertion

Ref	Expression
caovass	|- ( ( 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   x, F, y, z

Proof of Theorem caovass

Step	Hyp	Ref	Expression
1		caovass.1	. 2 |- A e. _V
2		caovass.2	. 2 |- B e. _V
3		caovass.3	. 2 |- C e. _V
4		tru 1399	. . 3 |- T.
5		caovass.4	. . . . 5 |- ( ( x F y ) F z ) = ( x F ( y F z ) )
6	5	a1i 9	. . . 4 |- ( ( T. /\ ( x e. _V /\ y e. _V /\ z e. _V ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
7	6	caovassg 6164	. . 3 |- ( ( T. /\ ( A e. _V /\ B e. _V /\ C e. _V ) ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
8	4, 7	mpan 424	. 2 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9	1, 2, 3, 8	mp3an 1371	1 |- ( ( A F B ) F C ) = ( A F ( B F C ) )

Syntax hints:   /\ wa 104   /\ w3a 1002   = wceq 1395   T. wtru 1396   e. wcel 2200   _Vcvv 2799  (class class class)co 6001

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004

This theorem is referenced by:  caov32  6193  caov12  6194  caov31  6195  caov13  6196