Theorem caovass 6037

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 1357	. . 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 6035	. . 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 1337	1 |- ( ( A F B ) F C ) = ( A F ( B F C ) )

Syntax hints:   /\ wa 104   /\ w3a 978   = wceq 1353   T. wtru 1354   e. wcel 2148   _Vcvv 2739  (class class class)co 5877

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by:  caov32  6064  caov12  6065  caov31  6066  caov13  6067