Theorem rexun 3443

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

Assertion

Ref	Expression
rexun	|- (E.x e. (A u. B)ph <-> (E.x e. A ph \/ E.x e. B ph))

Proof of Theorem rexun

Step	Hyp	Ref	Expression
1		df-rex 2620	. 2 |- (E.x e. (A u. B)ph <-> E.x(x e. (A u. B) /\ ph))
2		19.43 1605	. . 3 |- (E.x((x e. A /\ ph) \/ (x e. B /\ ph)) <-> (E.x(x e. A /\ ph) \/ E.x(x e. B /\ ph)))
3		elun 3220	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
4	3	anbi1i 676	. . . . 5 |- ((x e. (A u. B) /\ ph) <-> ((x e. A \/ x e. B) /\ ph))
5		andir 838	. . . . 5 |- (((x e. A \/ x e. B) /\ ph) <-> ((x e. A /\ ph) \/ (x e. B /\ ph)))
6	4, 5	bitri 240	. . . 4 |- ((x e. (A u. B) /\ ph) <-> ((x e. A /\ ph) \/ (x e. B /\ ph)))
7	6	exbii 1582	. . 3 |- (E.x(x e. (A u. B) /\ ph) <-> E.x((x e. A /\ ph) \/ (x e. B /\ ph)))
8		df-rex 2620	. . . 4 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
9		df-rex 2620	. . . 4 |- (E.x e. B ph <-> E.x(x e. B /\ ph))
10	8, 9	orbi12i 507	. . 3 |- ((E.x e. A ph \/ E.x e. B ph) <-> (E.x(x e. A /\ ph) \/ E.x(x e. B /\ ph)))
11	2, 7, 10	3bitr4i 268	. 2 |- (E.x(x e. (A u. B) /\ ph) <-> (E.x e. A ph \/ E.x e. B ph))
12	1, 11	bitri 240	1 |- (E.x e. (A u. B)ph <-> (E.x e. A ph \/ E.x e. B ph))

Syntax hints:   <-> wb 176   \/ wo 357   /\ wa 358  E.wex 1541   e. wcel 1710  E.wrex 2615   u. cun 3207

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 2478  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214

This theorem is referenced by:  rexprg  3776  rextpg  3778  iunxun  4047  pw1un  4163  nnadjoin  4520  tfinnn  4534  phiun  4614