Theorem caovdi 6098

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 1368	. . 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 6093	. . 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 424	. 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 1348	1 |- ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) )

Syntax hints:   /\ wa 104   /\ w3a 980   = wceq 1364   T. wtru 1365   e. wcel 2164   _Vcvv 2760  (class class class)co 5918

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by: (None)