Theorem rexun 3444

Description: Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)

Assertion

Ref	Expression
rexun	⊢ (∃x ∈ (A ∪ B)φ ↔ (∃x ∈ A φ ∨ ∃x ∈ B φ))

Proof of Theorem rexun

Step	Hyp	Ref	Expression
1		df-rex 2621	. 2 ⊢ (∃x ∈ (A ∪ B)φ ↔ ∃x(x ∈ (A ∪ B) ∧ φ))
2		19.43 1605	. . 3 ⊢ (∃x((x ∈ A ∧ φ) ∨ (x ∈ B ∧ φ)) ↔ (∃x(x ∈ A ∧ φ) ∨ ∃x(x ∈ B ∧ φ)))
3		elun 3221	. . . . . 6 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
4	3	anbi1i 676	. . . . 5 ⊢ ((x ∈ (A ∪ B) ∧ φ) ↔ ((x ∈ A ∨ x ∈ B) ∧ φ))
5		andir 838	. . . . 5 ⊢ (((x ∈ A ∨ x ∈ B) ∧ φ) ↔ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ φ)))
6	4, 5	bitri 240	. . . 4 ⊢ ((x ∈ (A ∪ B) ∧ φ) ↔ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ φ)))
7	6	exbii 1582	. . 3 ⊢ (∃x(x ∈ (A ∪ B) ∧ φ) ↔ ∃x((x ∈ A ∧ φ) ∨ (x ∈ B ∧ φ)))
8		df-rex 2621	. . . 4 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
9		df-rex 2621	. . . 4 ⊢ (∃x ∈ B φ ↔ ∃x(x ∈ B ∧ φ))
10	8, 9	orbi12i 507	. . 3 ⊢ ((∃x ∈ A φ ∨ ∃x ∈ B φ) ↔ (∃x(x ∈ A ∧ φ) ∨ ∃x(x ∈ B ∧ φ)))
11	2, 7, 10	3bitr4i 268	. 2 ⊢ (∃x(x ∈ (A ∪ B) ∧ φ) ↔ (∃x ∈ A φ ∨ ∃x ∈ B φ))
12	1, 11	bitri 240	1 ⊢ (∃x ∈ (A ∪ B)φ ↔ (∃x ∈ A φ ∨ ∃x ∈ B φ))

Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃wrex 2616   ∪ 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-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215

This theorem is referenced by:  rexprg  3777  rextpg  3779  iunxun  4048  pw1un  4164  nnadjoin  4521  tfinnn  4535  phiun  4615