MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  indifdi Structured version   Visualization version   GIF version

Theorem indifdi 4282
Description: Distribute intersection over difference. (Contributed by BTernaryTau, 14-Aug-2024.)
Assertion
Ref Expression
indifdi (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐴𝐶))

Proof of Theorem indifdi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elin 3963 . . 3 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
2 eldif 3957 . . . 4 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
32anbi2i 621 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4 abai 823 . . . . 5 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶)))
54anbi2i 621 . . . 4 ((𝑥𝐵 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶))))
6 an12 641 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
7 eldif 3957 . . . . 5 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐴𝐶)))
8 elin 3963 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
98bicomi 223 . . . . . 6 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴𝐵))
10 imnan 398 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐶) ↔ ¬ (𝑥𝐴𝑥𝐶))
11 elin 3963 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
1210, 11xchbinxr 334 . . . . . 6 ((𝑥𝐴 → ¬ 𝑥𝐶) ↔ ¬ 𝑥 ∈ (𝐴𝐶))
139, 12anbi12i 625 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → ¬ 𝑥𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐴𝐶)))
14 an21 640 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → ¬ 𝑥𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶))))
157, 13, 143bitr2i 298 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶))))
165, 6, 153bitr4i 302 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)))
171, 3, 163bitri 296 . 2 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)))
1817eqriv 2727 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1539  wcel 2104  cdif 3944  cin 3946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3950  df-in 3954
This theorem is referenced by:  indifdir  4283  resdifdi  6234  iscnrm3rlem4  47663
  Copyright terms: Public domain W3C validator