Theorem difindir 3219

Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
difindir	|- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) )

Proof of Theorem difindir

Step	Hyp	Ref	Expression
1		inindir 3184	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) )
2		invdif 3206	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i B ) \ C )
3		invdif 3206	. . 3 |- ( A i^i ( _V \ C ) ) = ( A \ C )
4		invdif 3206	. . 3 |- ( B i^i ( _V \ C ) ) = ( B \ C )
5	3, 4	ineq12i 3165	. 2 |- ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) ) = ( ( A \ C ) i^i ( B \ C ) )
6	1, 2, 5	3eqtr3i 2109	1 |- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) )

Syntax hints:   = wceq 1284   _Vcvv 2601   \ cdif 2970   i^i cin 2972

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-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-in 2979

This theorem is referenced by: (None)