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Theorem undif4 3284
 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 pm2.621 666 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴 ∨ ¬ 𝑥𝐶) → ¬ 𝑥𝐶))
2 olc 632 . . . . . . 7 𝑥𝐶 → (𝑥𝐴 ∨ ¬ 𝑥𝐶))
31, 2impbid1 130 . . . . . 6 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴 ∨ ¬ 𝑥𝐶) ↔ ¬ 𝑥𝐶))
43anbi2d 437 . . . . 5 ((𝑥𝐴 → ¬ 𝑥𝐶) → (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶)))
5 eldif 2927 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
65orbi2i 679 . . . . . 6 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
7 ordi 729 . . . . . 6 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
86, 7bitri 173 . . . . 5 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
9 elun 3084 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
109anbi1i 431 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶))
114, 8, 103bitr4g 212 . . . 4 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶)))
12 elun 3084 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
13 eldif 2927 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶))
1411, 12, 133bitr4g 212 . . 3 ((𝑥𝐴 → ¬ 𝑥𝐶) → (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
1514alimi 1344 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝑥𝐶) → ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
16 disj1 3270 . 2 ((𝐴𝐶) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐶))
17 dfcleq 2034 . 2 ((𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
1815, 16, 173imtr4i 190 1 ((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  ∀wal 1241   = wceq 1243   ∈ wcel 1393   ∖ cdif 2914   ∪ cun 2915   ∩ cin 2916  ∅c0 3224 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-nul 3225 This theorem is referenced by:  phplem1  6315
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