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Theorem indifdirOLD 4284
Description: Obsolete version of indifdir 4283 as of 14-Aug-2024. (Contributed by Scott Fenton, 14-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
indifdirOLD ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Proof of Theorem indifdirOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm3.24 404 . . . . . . . 8 ¬ (𝑥𝐶 ∧ ¬ 𝑥𝐶)
21intnan 488 . . . . . . 7 ¬ (𝑥𝐴 ∧ (𝑥𝐶 ∧ ¬ 𝑥𝐶))
3 anass 470 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐶 ∧ ¬ 𝑥𝐶)))
42, 3mtbir 323 . . . . . 6 ¬ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶)
54biorfi 938 . . . . 5 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ↔ (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶)))
6 an32 645 . . . . 5 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵))
7 andi 1007 . . . . 5 (((𝑥𝐴𝑥𝐶) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) ↔ (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶)))
85, 6, 73bitr4i 303 . . . 4 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)))
9 ianor 981 . . . . 5 (¬ (𝑥𝐵𝑥𝐶) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶))
109anbi2i 624 . . . 4 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)))
118, 10bitr4i 278 . . 3 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
12 elin 3963 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶))
13 eldif 3957 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1413anbi1i 625 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
1512, 14bitri 275 . . 3 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
16 eldif 3957 . . . 4 (𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
17 elin 3963 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
18 elin 3963 . . . . . 6 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1918notbii 320 . . . . 5 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
2017, 19anbi12i 628 . . . 4 ((𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
2116, 20bitri 275 . . 3 (𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
2211, 15, 213bitr4i 303 . 2 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ 𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)))
2322eqriv 2730 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wo 846   = wceq 1542  wcel 2107  cdif 3944  cin 3946
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-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3950  df-in 3954
This theorem is referenced by: (None)
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