Theorem indifdir 3512

Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)

Assertion

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

Proof of Theorem indifdir

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm3.24 852	. . . . . . . 8 ⊢ ¬ (x ∈ C ∧ ¬ x ∈ C)
2	1	intnan 880	. . . . . . 7 ⊢ ¬ (x ∈ A ∧ (x ∈ C ∧ ¬ x ∈ C))
3		anass 630	. . . . . . 7 ⊢ (((x ∈ A ∧ x ∈ C) ∧ ¬ x ∈ C) ↔ (x ∈ A ∧ (x ∈ C ∧ ¬ x ∈ C)))
4	2, 3	mtbir 290	. . . . . 6 ⊢ ¬ ((x ∈ A ∧ x ∈ C) ∧ ¬ x ∈ C)
5	4	biorfi 396	. . . . 5 ⊢ (((x ∈ A ∧ x ∈ C) ∧ ¬ x ∈ B) ↔ (((x ∈ A ∧ x ∈ C) ∧ ¬ x ∈ B) ∨ ((x ∈ A ∧ x ∈ C) ∧ ¬ x ∈ C)))
6		an32 773	. . . . 5 ⊢ (((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C) ↔ ((x ∈ A ∧ x ∈ C) ∧ ¬ x ∈ B))
7		andi 837	. . . . 5 ⊢ (((x ∈ A ∧ x ∈ C) ∧ (¬ x ∈ B ∨ ¬ x ∈ C)) ↔ (((x ∈ A ∧ x ∈ C) ∧ ¬ x ∈ B) ∨ ((x ∈ A ∧ x ∈ C) ∧ ¬ x ∈ C)))
8	5, 6, 7	3bitr4i 268	. . . 4 ⊢ (((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C) ↔ ((x ∈ A ∧ x ∈ C) ∧ (¬ x ∈ B ∨ ¬ x ∈ C)))
9		ianor 474	. . . . 5 ⊢ (¬ (x ∈ B ∧ x ∈ C) ↔ (¬ x ∈ B ∨ ¬ x ∈ C))
10	9	anbi2i 675	. . . 4 ⊢ (((x ∈ A ∧ x ∈ C) ∧ ¬ (x ∈ B ∧ x ∈ C)) ↔ ((x ∈ A ∧ x ∈ C) ∧ (¬ x ∈ B ∨ ¬ x ∈ C)))
11	8, 10	bitr4i 243	. . 3 ⊢ (((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C) ↔ ((x ∈ A ∧ x ∈ C) ∧ ¬ (x ∈ B ∧ x ∈ C)))
12		elin 3220	. . . 4 ⊢ (x ∈ ((A ∖ B) ∩ C) ↔ (x ∈ (A ∖ B) ∧ x ∈ C))
13		eldif 3222	. . . . 5 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
14	13	anbi1i 676	. . . 4 ⊢ ((x ∈ (A ∖ B) ∧ x ∈ C) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C))
15	12, 14	bitri 240	. . 3 ⊢ (x ∈ ((A ∖ B) ∩ C) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C))
16		eldif 3222	. . . 4 ⊢ (x ∈ ((A ∩ C) ∖ (B ∩ C)) ↔ (x ∈ (A ∩ C) ∧ ¬ x ∈ (B ∩ C)))
17		elin 3220	. . . . 5 ⊢ (x ∈ (A ∩ C) ↔ (x ∈ A ∧ x ∈ C))
18		elin 3220	. . . . . 6 ⊢ (x ∈ (B ∩ C) ↔ (x ∈ B ∧ x ∈ C))
19	18	notbii 287	. . . . 5 ⊢ (¬ x ∈ (B ∩ C) ↔ ¬ (x ∈ B ∧ x ∈ C))
20	17, 19	anbi12i 678	. . . 4 ⊢ ((x ∈ (A ∩ C) ∧ ¬ x ∈ (B ∩ C)) ↔ ((x ∈ A ∧ x ∈ C) ∧ ¬ (x ∈ B ∧ x ∈ C)))
21	16, 20	bitri 240	. . 3 ⊢ (x ∈ ((A ∩ C) ∖ (B ∩ C)) ↔ ((x ∈ A ∧ x ∈ C) ∧ ¬ (x ∈ B ∧ x ∈ C)))
22	11, 15, 21	3bitr4i 268	. 2 ⊢ (x ∈ ((A ∖ B) ∩ C) ↔ x ∈ ((A ∩ C) ∖ (B ∩ C)))
23	22	eqriv 2350	1 ⊢ ((A ∖ B) ∩ C) = ((A ∩ C) ∖ (B ∩ C))

Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3207   ∩ 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-dif 3216

This theorem is referenced by: (None)