Theorem undif4 3471

Description: Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
undif4	⊢ ((𝐴 ∩ 𝐶) = ∅ → (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶))

Proof of Theorem undif4

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2.621 737	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶) → ¬ 𝑥 ∈ 𝐶))
2		olc 701	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶))
3	1, 2	impbid1 141	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶) ↔ ¬ 𝑥 ∈ 𝐶))
4	3	anbi2d 460	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶)))
5		eldif 3125	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
6	5	orbi2i 752	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
7		ordi 806	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶)))
8	6, 7	bitri 183	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶)))
9		elun 3263	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
10	9	anbi1i 454	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶))
11	4, 8, 10	3bitr4g 222	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶)))
12		elun 3263	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)))
13		eldif 3125	. . . 4 ⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ 𝐶) ↔ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶))
14	11, 12, 13	3bitr4g 222	. . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ 𝐶)))
15	14	alimi 1443	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ 𝐶)))
16		disj1 3459	. 2 ⊢ ((𝐴 ∩ 𝐶) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶))
17		dfcleq 2159	. 2 ⊢ ((𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ 𝐶)))
18	15, 16, 17	3imtr4i 200	1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  ∀wal 1341   = wceq 1343   ∈ wcel 2136   ∖ cdif 3113   ∪ cun 3114   ∩ cin 3115  ∅c0 3409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-nul 3410

This theorem is referenced by:  phplem1  6818