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Theorem undif4 3987
 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 pm2.621 423 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴 ∨ ¬ 𝑥𝐶) → ¬ 𝑥𝐶))
2 olc 398 . . . . . . 7 𝑥𝐶 → (𝑥𝐴 ∨ ¬ 𝑥𝐶))
31, 2impbid1 214 . . . . . 6 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴 ∨ ¬ 𝑥𝐶) ↔ ¬ 𝑥𝐶))
43anbi2d 736 . . . . 5 ((𝑥𝐴 → ¬ 𝑥𝐶) → (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶)))
5 eldif 3550 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
65orbi2i 540 . . . . . 6 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
7 ordi 904 . . . . . 6 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
86, 7bitri 263 . . . . 5 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
9 elun 3715 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
109anbi1i 727 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶))
114, 8, 103bitr4g 302 . . . 4 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶)))
12 elun 3715 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
13 eldif 3550 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶))
1411, 12, 133bitr4g 302 . . 3 ((𝑥𝐴 → ¬ 𝑥𝐶) → (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
1514alimi 1730 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝑥𝐶) → ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
16 disj1 3971 . 2 ((𝐴𝐶) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐶))
17 dfcleq 2604 . 2 ((𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
1815, 16, 173imtr4i 280 1 ((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-nul 3875 This theorem is referenced by:  phplem1  8024  infdifsn  8437  difico  28935  caratheodorylem1  39416
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