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Theorem undif4 3342
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 pm2.621 701 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴 ∨ ¬ 𝑥𝐶) → ¬ 𝑥𝐶))
2 olc 667 . . . . . . 7 𝑥𝐶 → (𝑥𝐴 ∨ ¬ 𝑥𝐶))
31, 2impbid1 140 . . . . . 6 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴 ∨ ¬ 𝑥𝐶) ↔ ¬ 𝑥𝐶))
43anbi2d 452 . . . . 5 ((𝑥𝐴 → ¬ 𝑥𝐶) → (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶)))
5 eldif 3006 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
65orbi2i 714 . . . . . 6 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
7 ordi 765 . . . . . 6 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
86, 7bitri 182 . . . . 5 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
9 elun 3139 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
109anbi1i 446 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶))
114, 8, 103bitr4g 221 . . . 4 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶)))
12 elun 3139 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
13 eldif 3006 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶))
1411, 12, 133bitr4g 221 . . 3 ((𝑥𝐴 → ¬ 𝑥𝐶) → (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
1514alimi 1389 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝑥𝐶) → ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
16 disj1 3330 . 2 ((𝐴𝐶) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐶))
17 dfcleq 2082 . 2 ((𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
1815, 16, 173imtr4i 199 1 ((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664  wal 1287   = wceq 1289  wcel 1438  cdif 2994  cun 2995  cin 2996  c0 3284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-nul 3285
This theorem is referenced by:  phplem1  6548
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