Theorem caovass 6360

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 1374	. . 3 |- T.
5		caovass.4	. . . . 5 |- ( ( x F y ) F z ) = ( x F ( y F z ) )
6	5	a1i 11	. . . 4 |- ( ( T. /\ ( x e. _V /\ y e. _V /\ z e. _V ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
7	6	caovassg 6358	. . 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 670	. 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 1315	1 |- ( ( A F B ) F C ) = ( A F ( B F C ) )

Syntax hints:   /\ wa 369   /\ w3a 965   = wceq 1370   T. wtru 1371   e. wcel 1758   _Vcvv 3065  (class class class)co 6187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-iota 5476  df-fv 5521  df-ov 6190

This theorem is referenced by:  caov32  6387  caov12  6388  caov31  6389  caov13  6390  caov4  6391  caov411  6392  caovdilem  6395  caovmo  6397