Theorem indifdi 4240

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 3915	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶)))
2		eldif 3909	. . . 4 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
3	2	anbi2i 635	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
4		abai 839	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶)))
5	4	anbi2i 635	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶))))
6		an12 658	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
7		eldif 3909	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐶)))
8		elin 3915	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
9	8	bicomi 227	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐴 ∩ 𝐵))
10		imnan 405	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
11		elin 3915	. . . . . . 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 640	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐶)))
14		an21 657	. . . . 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 2758	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896   ∩ cin 3898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-in 3906

This theorem is used by:  indifdir  4241  resdifdi  6237  iscnrm3rlem4  50050