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Theorem undif4 4412
Description: Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
undif4 ((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))

Proof of Theorem undif4
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm2.621 892 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴 ∨ ¬ 𝑥𝐶) → ¬ 𝑥𝐶))
2 olc 862 . . . . . . 7 𝑥𝐶 → (𝑥𝐴 ∨ ¬ 𝑥𝐶))
31, 2impbid1 226 . . . . . 6 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴 ∨ ¬ 𝑥𝐶) ↔ ¬ 𝑥𝐶))
43anbi2d 628 . . . . 5 ((𝑥𝐴 → ¬ 𝑥𝐶) → (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶)))
5 eldif 3943 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
65orbi2i 906 . . . . . 6 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
7 ordi 999 . . . . . 6 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
86, 7bitri 276 . . . . 5 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
9 elun 4122 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
109anbi1i 623 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶))
114, 8, 103bitr4g 315 . . . 4 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶)))
12 elun 4122 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
13 eldif 3943 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶))
1411, 12, 133bitr4g 315 . . 3 ((𝑥𝐴 → ¬ 𝑥𝐶) → (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
1514alimi 1803 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝑥𝐶) → ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
16 disj1 4397 . 2 ((𝐴𝐶) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐶))
17 dfcleq 2812 . 2 ((𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
1815, 16, 173imtr4i 293 1 ((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  wal 1526   = wceq 1528  wcel 2105  cdif 3930  cun 3931  cin 3932  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-nul 4289
This theorem is referenced by:  phplem1  8684  infdifsn  9108  difico  30432  lindsunlem  30919  caratheodorylem1  42685
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