Theorem ralunb 3444

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	|- (A.x e. (A u. B)ph <-> (A.x e. A ph /\ A.x e. B ph))

Proof of Theorem ralunb

Step	Hyp	Ref	Expression
1		elun 3220	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
2	1	imbi1i 315	. . . . 5 |- ((x e. (A u. B) -> ph) <-> ((x e. A \/ x e. B) -> ph))
3		jaob 758	. . . . 5 |- (((x e. A \/ x e. B) -> ph) <-> ((x e. A -> ph) /\ (x e. B -> ph)))
4	2, 3	bitri 240	. . . 4 |- ((x e. (A u. B) -> ph) <-> ((x e. A -> ph) /\ (x e. B -> ph)))
5	4	albii 1566	. . 3 |- (A.x(x e. (A u. B) -> ph) <-> A.x((x e. A -> ph) /\ (x e. B -> ph)))
6		19.26 1593	. . 3 |- (A.x((x e. A -> ph) /\ (x e. B -> ph)) <-> (A.x(x e. A -> ph) /\ A.x(x e. B -> ph)))
7	5, 6	bitri 240	. 2 |- (A.x(x e. (A u. B) -> ph) <-> (A.x(x e. A -> ph) /\ A.x(x e. B -> ph)))
8		df-ral 2619	. 2 |- (A.x e. (A u. B)ph <-> A.x(x e. (A u. B) -> ph))
9		df-ral 2619	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
10		df-ral 2619	. . 3 |- (A.x e. B ph <-> A.x(x e. B -> ph))
11	9, 10	anbi12i 678	. 2 |- ((A.x e. A ph /\ A.x e. B ph) <-> (A.x(x e. A -> ph) /\ A.x(x e. B -> ph)))
12	7, 8, 11	3bitr4i 268	1 |- (A.x e. (A u. B)ph <-> (A.x e. A ph /\ A.x e. B ph))

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358  A.wal 1540   e. wcel 1710  A.wral 2614   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-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