Theorem indifdi 4217

Description: Distribute intersection over difference. (Contributed by BTernaryTau, 14-Aug-2024.)

Assertion

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

Proof of Theorem indifdi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3903	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶)))
2		eldif 3897	. . . 4 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
3	2	anbi2i 623	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
4		abai 824	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶)))
5	4	anbi2i 623	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶))))
6		an12 642	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
7		eldif 3897	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐶)))
8		elin 3903	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
9	8	bicomi 223	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐴 ∩ 𝐵))
10		imnan 400	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
11		elin 3903	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
12	10, 11	xchbinxr 335	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) ↔ ¬ 𝑥 ∈ (𝐴 ∩ 𝐶))
13	9, 12	anbi12i 627	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐶)))
14		an21 641	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶))))
15	7, 13, 14	3bitr2i 299	. . . 4 ⊢ (𝑥 ∈ ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶))))
16	5, 6, 15	3bitr4i 303	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶)))
17	1, 3, 16	3bitri 297	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶)))
18	17	eqriv 2735	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   ∩ cin 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894

This theorem is referenced by:  indifdir  4218  resdifdi  6139  iscnrm3rlem4  46237