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Theorem undif4 3260
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 pm2.621 653 . . . . . . 7 ((x A → ¬ x 𝐶) → ((x A ¬ x 𝐶) → ¬ x 𝐶))
2 olc 619 . . . . . . 7 x 𝐶 → (x A ¬ x 𝐶))
31, 2impbid1 130 . . . . . 6 ((x A → ¬ x 𝐶) → ((x A ¬ x 𝐶) ↔ ¬ x 𝐶))
43anbi2d 440 . . . . 5 ((x A → ¬ x 𝐶) → (((x A x B) (x A ¬ x 𝐶)) ↔ ((x A x B) ¬ x 𝐶)))
5 eldif 2903 . . . . . . 7 (x (B𝐶) ↔ (x B ¬ x 𝐶))
65orbi2i 666 . . . . . 6 ((x A x (B𝐶)) ↔ (x A (x B ¬ x 𝐶)))
7 ordi 717 . . . . . 6 ((x A (x B ¬ x 𝐶)) ↔ ((x A x B) (x A ¬ x 𝐶)))
86, 7bitri 173 . . . . 5 ((x A x (B𝐶)) ↔ ((x A x B) (x A ¬ x 𝐶)))
9 elun 3060 . . . . . 6 (x (AB) ↔ (x A x B))
109anbi1i 434 . . . . 5 ((x (AB) ¬ x 𝐶) ↔ ((x A x B) ¬ x 𝐶))
114, 8, 103bitr4g 212 . . . 4 ((x A → ¬ x 𝐶) → ((x A x (B𝐶)) ↔ (x (AB) ¬ x 𝐶)))
12 elun 3060 . . . 4 (x (A ∪ (B𝐶)) ↔ (x A x (B𝐶)))
13 eldif 2903 . . . 4 (x ((AB) ∖ 𝐶) ↔ (x (AB) ¬ x 𝐶))
1411, 12, 133bitr4g 212 . . 3 ((x A → ¬ x 𝐶) → (x (A ∪ (B𝐶)) ↔ x ((AB) ∖ 𝐶)))
1514alimi 1324 . 2 (x(x A → ¬ x 𝐶) → x(x (A ∪ (B𝐶)) ↔ x ((AB) ∖ 𝐶)))
16 disj1 3246 . 2 ((A𝐶) = ∅ ↔ x(x A → ¬ x 𝐶))
17 dfcleq 2017 . 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 616  wal 1226   = wceq 1228   wcel 1375  cdif 2890  cun 2891  cin 2892  c0 3200
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-v 2536  df-dif 2896  df-un 2898  df-in 2900  df-nul 3201
This theorem is referenced by: (None)
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