Theorem difundir 3412

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

Assertion

Ref	Expression
difundir	|- ( ( A u. B ) \ C ) = ( ( A \ C ) u. ( B \ C ) )

Proof of Theorem difundir

Step	Hyp	Ref	Expression
1		indir 3408	. 2 |- ( ( A u. B ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ C ) ) u. ( B i^i ( _V \ C ) ) )
2		invdif 3401	. 2 |- ( ( A u. B ) i^i ( _V \ C ) ) = ( ( A u. B ) \ C )
3		invdif 3401	. . 3 |- ( A i^i ( _V \ C ) ) = ( A \ C )
4		invdif 3401	. . 3 |- ( B i^i ( _V \ C ) ) = ( B \ C )
5	3, 4	uneq12i 3311	. 2 |- ( ( A i^i ( _V \ C ) ) u. ( B i^i ( _V \ C ) ) ) = ( ( A \ C ) u. ( B \ C ) )
6	1, 2, 5	3eqtr3i 2222	1 |- ( ( A u. B ) \ C ) = ( ( A \ C ) u. ( B \ C ) )

Syntax hints:   = wceq 1364   _Vcvv 2760   \ cdif 3150   u. cun 3151   i^i cin 3152

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-in1 615  ax-in2 616  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-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-un 3157  df-in 3159

This theorem is referenced by:  symdif1  3424  difun2  3526  diftpsn3  3759  unfiin  6982  setsfun0  12654  strleund  12721  strleun  12722