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Theorem indifdi 4175
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 3860 . . 3 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
2 eldif 3854 . . . 4 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
32anbi2i 626 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4 abai 826 . . . . 5 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶)))
54anbi2i 626 . . . 4 ((𝑥𝐵 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶))))
6 an12 645 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
7 eldif 3854 . . . . 5 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐴𝐶)))
8 elin 3860 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
98bicomi 227 . . . . . 6 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴𝐵))
10 imnan 403 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐶) ↔ ¬ (𝑥𝐴𝑥𝐶))
11 elin 3860 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
1210, 11xchbinxr 338 . . . . . 6 ((𝑥𝐴 → ¬ 𝑥𝐶) ↔ ¬ 𝑥 ∈ (𝐴𝐶))
139, 12anbi12i 630 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → ¬ 𝑥𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐴𝐶)))
14 an21 644 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → ¬ 𝑥𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶))))
157, 13, 143bitr2i 302 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶))))
165, 6, 153bitr4i 306 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)))
171, 3, 163bitri 300 . 2 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)))
1817eqriv 2735 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wcel 2113  cdif 3841  cin 3843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-dif 3847  df-in 3851
This theorem is referenced by:  indifdir  4176  resdifdi  6069  iscnrm3rlem4  45751
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