Theorem ralunb 3445

Description: Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
ralunb	⊢ (∀x ∈ (A ∪ B)φ ↔ (∀x ∈ A φ ∧ ∀x ∈ B φ))

Proof of Theorem ralunb

Step	Hyp	Ref	Expression
1		elun 3221	. . . . . 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 2620	. 2 ⊢ (∀x ∈ (A ∪ B)φ ↔ ∀x(x ∈ (A ∪ B) → φ))
9		df-ral 2620	. . 3 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
10		df-ral 2620	. . 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))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  ∀wral 2615   ∪ cun 3208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215

This theorem is referenced by:  ralun  3446  ralprg  3776  raltpg  3778  ralunsn  3880  ssofss  4077