Theorem difindir 3750

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 3702	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) )
2		invdif 3736	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i B ) \ C )
3		invdif 3736	. . 3 |- ( A i^i ( _V \ C ) ) = ( A \ C )
4		invdif 3736	. . 3 |- ( B i^i ( _V \ C ) ) = ( B \ C )
5	3, 4	ineq12i 3684	. 2 |- ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) ) = ( ( A \ C ) i^i ( B \ C ) )
6	1, 2, 5	3eqtr3i 2491	1 |- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) )

Syntax hints:   = wceq 1398   _Vcvv 3106   \ cdif 3458   i^i cin 3460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-in 3468

This theorem is referenced by:  ablfac1eulem  17321  ballotlemgun  28730