Theorem caovass 5805

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 1293	. . 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 5803	. . 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 415	. 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 1273	1 |- ( ( A F B ) F C ) = ( A F ( B F C ) )

Syntax hints:   /\ wa 102   /\ w3a 924   = wceq 1289   T. wtru 1290   e. wcel 1438   _Vcvv 2619  (class class class)co 5652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655

This theorem is referenced by:  caov32  5832  caov12  5833  caov31  5834  caov13  5835