Theorem caovdi 5680

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 1247	. . 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 5675	. . 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 400	. 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 1232	1 |- ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) )

Syntax hints:   /\ wa 97   /\ w3a 885   = wceq 1243   T. wtru 1244   e. wcel 1393   _Vcvv 2557  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by: (None)