Theorem difindir 3511

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

Assertion

Ref	Expression
difindir	⊢ ((A ∩ B) ∖ C) = ((A ∖ C) ∩ (B ∖ C))

Proof of Theorem difindir

Step	Hyp	Ref	Expression
1		inindir 3474	. 2 ⊢ ((A ∩ B) ∩ (V ∖ C)) = ((A ∩ (V ∖ C)) ∩ (B ∩ (V ∖ C)))
2		invdif 3497	. 2 ⊢ ((A ∩ B) ∩ (V ∖ C)) = ((A ∩ B) ∖ C)
3		invdif 3497	. . 3 ⊢ (A ∩ (V ∖ C)) = (A ∖ C)
4		invdif 3497	. . 3 ⊢ (B ∩ (V ∖ C)) = (B ∖ C)
5	3, 4	ineq12i 3456	. 2 ⊢ ((A ∩ (V ∖ C)) ∩ (B ∩ (V ∖ C))) = ((A ∖ C) ∩ (B ∖ C))
6	1, 2, 5	3eqtr3i 2381	1 ⊢ ((A ∩ B) ∖ C) = ((A ∖ C) ∩ (B ∖ C))

This proof depends on syntax axioms:   = wceq 1642  Vcvv 2860   ∖ cdif 3207   ∩ cin 3209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is used by: (None)