Theorem indifdi 4255

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 3929	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶)))
2		eldif 3923	. . . 4 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
3	2	anbi2i 634	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
4		abai 838	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶)))
5	4	anbi2i 634	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶))))
6		an12 657	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
7		eldif 3923	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐶)))
8		elin 3929	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
9	8	bicomi 227	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐴 ∩ 𝐵))
10		imnan 404	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
11		elin 3929	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
12	10, 11	xchbinxr 338	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) ↔ ¬ 𝑥 ∈ (𝐴 ∩ 𝐶))
13	9, 12	anbi12i 639	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐶)))
14		an21 656	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶))))
15	7, 13, 14	3bitr2i 302	. . . 4 ⊢ (𝑥 ∈ ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶))))
16	5, 6, 15	3bitr4i 306	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶)))
17	1, 3, 16	3bitri 300	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶)))
18	17	eqriv 2766	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∖ cdif 3910   ∩ cin 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-in 3920

This theorem is referenced by:  indifdir  4256  resdifdi  6238  iscnrm3rlem4  49606