Theorem difindir 3392

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 3355	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) )
2		invdif 3379	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i B ) \ C )
3		invdif 3379	. . 3 |- ( A i^i ( _V \ C ) ) = ( A \ C )
4		invdif 3379	. . 3 |- ( B i^i ( _V \ C ) ) = ( B \ C )
5	3, 4	ineq12i 3336	. 2 |- ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) ) = ( ( A \ C ) i^i ( B \ C ) )
6	1, 2, 5	3eqtr3i 2206	1 |- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) )

Syntax hints:   = wceq 1353   _Vcvv 2739   \ cdif 3128   i^i cin 3130

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-in 3137

This theorem is referenced by: (None)