Theorem difindi 3510

Description: Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
difindi	⊢ (A ∖ (B ∩ C)) = ((A ∖ B) ∪ (A ∖ C))

Proof of Theorem difindi

Step	Hyp	Ref	Expression
1		dfin3 3495	. . 3 ⊢ (B ∩ C) = (V ∖ ((V ∖ B) ∪ (V ∖ C)))
2	1	difeq2i 3383	. 2 ⊢ (A ∖ (B ∩ C)) = (A ∖ (V ∖ ((V ∖ B) ∪ (V ∖ C))))
3		indi 3502	. . 3 ⊢ (A ∩ ((V ∖ B) ∪ (V ∖ C))) = ((A ∩ (V ∖ B)) ∪ (A ∩ (V ∖ C)))
4		dfin2 3492	. . 3 ⊢ (A ∩ ((V ∖ B) ∪ (V ∖ C))) = (A ∖ (V ∖ ((V ∖ B) ∪ (V ∖ C))))
5		invdif 3497	. . . 4 ⊢ (A ∩ (V ∖ B)) = (A ∖ B)
6		invdif 3497	. . . 4 ⊢ (A ∩ (V ∖ C)) = (A ∖ C)
7	5, 6	uneq12i 3417	. . 3 ⊢ ((A ∩ (V ∖ B)) ∪ (A ∩ (V ∖ C))) = ((A ∖ B) ∪ (A ∖ C))
8	3, 4, 7	3eqtr3i 2381	. 2 ⊢ (A ∖ (V ∖ ((V ∖ B) ∪ (V ∖ C)))) = ((A ∖ B) ∪ (A ∖ C))
9	2, 8	eqtri 2373	1 ⊢ (A ∖ (B ∩ C)) = ((A ∖ B) ∪ (A ∖ C))

Syntax hints:   = wceq 1642  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209

This theorem was proved from 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 theorem 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-un 3215  df-dif 3216

This theorem is referenced by:  indm  3514