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Theorem indifdir 4026
Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
indifdir ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Proof of Theorem indifdir
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm3.24 962 . . . . . . . 8 ¬ (𝑥𝐶 ∧ ¬ 𝑥𝐶)
21intnan 998 . . . . . . 7 ¬ (𝑥𝐴 ∧ (𝑥𝐶 ∧ ¬ 𝑥𝐶))
3 anass 684 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐶 ∧ ¬ 𝑥𝐶)))
42, 3mtbir 312 . . . . . 6 ¬ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶)
54biorfi 421 . . . . 5 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ↔ (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶)))
6 an32 874 . . . . 5 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵))
7 andi 947 . . . . 5 (((𝑥𝐴𝑥𝐶) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) ↔ (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶)))
85, 6, 73bitr4i 292 . . . 4 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)))
9 ianor 510 . . . . 5 (¬ (𝑥𝐵𝑥𝐶) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶))
109anbi2i 732 . . . 4 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)))
118, 10bitr4i 267 . . 3 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
12 elin 3939 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶))
13 eldif 3725 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1413anbi1i 733 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
1512, 14bitri 264 . . 3 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
16 eldif 3725 . . . 4 (𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
17 elin 3939 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
18 elin 3939 . . . . . 6 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1918notbii 309 . . . . 5 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
2017, 19anbi12i 735 . . . 4 ((𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
2116, 20bitri 264 . . 3 (𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
2211, 15, 213bitr4i 292 . 2 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ 𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)))
2322eqriv 2757 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382  wa 383   = wceq 1632  wcel 2139  cdif 3712  cin 3714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-in 3722
This theorem is referenced by:  preddif  5866  fresaun  6236  uniioombllem4  23554  subsalsal  41080
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