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Mirrors > Home > ILE Home > Th. List > ralunb | GIF version |
Description: Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
ralunb | ⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 3144 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
2 | 1 | imbi1i 237 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → 𝜑)) |
3 | jaob 667 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → 𝜑) ↔ ((𝑥 ∈ 𝐴 → 𝜑) ∧ (𝑥 ∈ 𝐵 → 𝜑))) | |
4 | 2, 3 | bitri 183 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ((𝑥 ∈ 𝐴 → 𝜑) ∧ (𝑥 ∈ 𝐵 → 𝜑))) |
5 | 4 | albii 1405 | . . 3 ⊢ (∀𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ∀𝑥((𝑥 ∈ 𝐴 → 𝜑) ∧ (𝑥 ∈ 𝐵 → 𝜑))) |
6 | 19.26 1416 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝜑) ∧ (𝑥 ∈ 𝐵 → 𝜑)) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))) | |
7 | 5, 6 | bitri 183 | . 2 ⊢ (∀𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))) |
8 | df-ral 2365 | . 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∀𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) → 𝜑)) | |
9 | df-ral 2365 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
10 | df-ral 2365 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑)) | |
11 | 9, 10 | anbi12i 449 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))) |
12 | 7, 8, 11 | 3bitr4i 211 | 1 ⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 665 ∀wal 1288 ∈ wcel 1439 ∀wral 2360 ∪ cun 3000 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-v 2624 df-un 3006 |
This theorem is referenced by: ralun 3185 ralprg 3499 raltpg 3501 ralunsn 3649 rexfiuz 10485 modfsummodlemstep 10914 modfsummod 10915 zsupcllemstep 11282 prmind2 11443 nninfsellemdc 12205 nninfsellemsuc 12207 |
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