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Theorem undif4 3281
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 ((x A → ¬ x 𝐶) → ((x A ¬ x 𝐶) → ¬ x 𝐶))
2 olc 632 . . . . . . 7 x 𝐶 → (x A ¬ x 𝐶))
31, 2impbid1 130 . . . . . 6 ((x A → ¬ x 𝐶) → ((x A ¬ x 𝐶) ↔ ¬ x 𝐶))
43anbi2d 437 . . . . 5 ((x A → ¬ x 𝐶) → (((x A x B) (x A ¬ x 𝐶)) ↔ ((x A x B) ¬ x 𝐶)))
5 eldif 2924 . . . . . . 7 (x (B𝐶) ↔ (x B ¬ x 𝐶))
65orbi2i 679 . . . . . 6 ((x A x (B𝐶)) ↔ (x A (x B ¬ x 𝐶)))
7 ordi 729 . . . . . 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 3081 . . . . . 6 (x (AB) ↔ (x A x B))
109anbi1i 431 . . . . 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 3081 . . . 4 (x (A ∪ (B𝐶)) ↔ (x A x (B𝐶)))
13 eldif 2924 . . . 4 (x ((AB) ∖ 𝐶) ↔ (x (AB) ¬ x 𝐶))
1411, 12, 133bitr4g 212 . . 3 ((x A → ¬ x 𝐶) → (x (A ∪ (B𝐶)) ↔ x ((AB) ∖ 𝐶)))
1514alimi 1344 . 2 (x(x A → ¬ x 𝐶) → x(x (A ∪ (B𝐶)) ↔ x ((AB) ∖ 𝐶)))
16 disj1 3267 . 2 ((A𝐶) = ∅ ↔ x(x A → ¬ x 𝐶))
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 2911  cun 2912  cin 2913  c0 3221
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 2308  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-nul 3222
This theorem is referenced by: (None)
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