Theorem caovdi 6032

Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)

Hypotheses

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

Assertion

Ref	Expression
caovdi	|- ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, F, y, z   x, G, y, z

Proof of Theorem caovdi

Step	Hyp	Ref	Expression
1		caovdi.1	. 2 |- A e. _V
2		caovdi.2	. 2 |- B e. _V
3		caovdi.3	. 2 |- C e. _V
4		tru 1352	. . 3 |- T.
5		caovdi.4	. . . . 5 |- ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) )
6	5	a1i 9	. . . 4 |- ( ( T. /\ ( x e. _V /\ y e. _V /\ z e. _V ) ) -> ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) )
7	6	caovdig 6027	. . 3 |- ( ( T. /\ ( A e. _V /\ B e. _V /\ C e. _V ) ) -> ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) ) )
8	4, 7	mpan 422	. 2 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) ) )
9	1, 2, 3, 8	mp3an 1332	1 |- ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) )

Syntax hints:   /\ wa 103   /\ w3a 973   = wceq 1348   T. wtru 1349   e. wcel 2141   _Vcvv 2730  (class class class)co 5853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by: (None)