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Theorem indifdi 4284
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 3965 . . 3 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
2 eldif 3959 . . . 4 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
32anbi2i 624 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4 abai 826 . . . . 5 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶)))
54anbi2i 624 . . . 4 ((𝑥𝐵 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶))))
6 an12 644 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
7 eldif 3959 . . . . 5 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐴𝐶)))
8 elin 3965 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
98bicomi 223 . . . . . 6 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴𝐵))
10 imnan 401 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐶) ↔ ¬ (𝑥𝐴𝑥𝐶))
11 elin 3965 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
1210, 11xchbinxr 335 . . . . . 6 ((𝑥𝐴 → ¬ 𝑥𝐶) ↔ ¬ 𝑥 ∈ (𝐴𝐶))
139, 12anbi12i 628 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → ¬ 𝑥𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐴𝐶)))
14 an21 643 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → ¬ 𝑥𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶))))
157, 13, 143bitr2i 299 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)) ↔ (𝑥𝐵 ∧ (𝑥𝐴 ∧ (𝑥𝐴 → ¬ 𝑥𝐶))))
165, 6, 153bitr4i 303 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)))
171, 3, 163bitri 297 . 2 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐴𝐶)))
1817eqriv 2730 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  cdif 3946  cin 3948
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-in 3956
This theorem is referenced by:  indifdir  4285  resdifdi  6236  iscnrm3rlem4  47576
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