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Mirrors > Home > NFE 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 3220 | . . . . . 6 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B)) | |
2 | 1 | imbi1i 315 | . . . . 5 ⊢ ((x ∈ (A ∪ B) → φ) ↔ ((x ∈ A ∨ x ∈ B) → φ)) |
3 | jaob 758 | . . . . 5 ⊢ (((x ∈ A ∨ x ∈ B) → φ) ↔ ((x ∈ A → φ) ∧ (x ∈ B → φ))) | |
4 | 2, 3 | bitri 240 | . . . 4 ⊢ ((x ∈ (A ∪ B) → φ) ↔ ((x ∈ A → φ) ∧ (x ∈ B → φ))) |
5 | 4 | albii 1566 | . . 3 ⊢ (∀x(x ∈ (A ∪ B) → φ) ↔ ∀x((x ∈ A → φ) ∧ (x ∈ B → φ))) |
6 | 19.26 1593 | . . 3 ⊢ (∀x((x ∈ A → φ) ∧ (x ∈ B → φ)) ↔ (∀x(x ∈ A → φ) ∧ ∀x(x ∈ B → φ))) | |
7 | 5, 6 | bitri 240 | . 2 ⊢ (∀x(x ∈ (A ∪ B) → φ) ↔ (∀x(x ∈ A → φ) ∧ ∀x(x ∈ B → φ))) |
8 | df-ral 2619 | . 2 ⊢ (∀x ∈ (A ∪ B)φ ↔ ∀x(x ∈ (A ∪ B) → φ)) | |
9 | df-ral 2619 | . . 3 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ)) | |
10 | df-ral 2619 | . . 3 ⊢ (∀x ∈ B φ ↔ ∀x(x ∈ B → φ)) | |
11 | 9, 10 | anbi12i 678 | . 2 ⊢ ((∀x ∈ A φ ∧ ∀x ∈ B φ) ↔ (∀x(x ∈ A → φ) ∧ ∀x(x ∈ B → φ))) |
12 | 7, 8, 11 | 3bitr4i 268 | 1 ⊢ (∀x ∈ (A ∪ B)φ ↔ (∀x ∈ A φ ∧ ∀x ∈ B φ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∨ wo 357 ∧ wa 358 ∀wal 1540 ∈ wcel 1710 ∀wral 2614 ∪ cun 3207 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 |
This theorem is referenced by: ralun 3445 ralprg 3775 raltpg 3777 ralunsn 3879 ssofss 4076 |
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