Theorem elunirab 3904

Description: Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)

Assertion

Ref	Expression
elunirab	|- (A e. U.{x e. B | ph} <-> E.x e. B (A e. x /\ ph))

Distinct variable group:   x,A

Allowed substitution hints:   ph(x)   B(x)

Proof of Theorem elunirab

Step	Hyp	Ref	Expression
1		eluniab 3903	. 2 |- (A e. U.{x | (x e. B /\ ph)} <-> E.x(A e. x /\ (x e. B /\ ph)))
2		df-rab 2623	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
3	2	unieqi 3901	. . 3 |- U.{x e. B | ph} = U.{x | (x e. B /\ ph)}
4	3	eleq2i 2417	. 2 |- (A e. U.{x e. B | ph} <-> A e. U.{x | (x e. B /\ ph)})
5		df-rex 2620	. . 3 |- (E.x e. B (A e. x /\ ph) <-> E.x(x e. B /\ (A e. x /\ ph)))
6		an12 772	. . . 4 |- ((x e. B /\ (A e. x /\ ph)) <-> (A e. x /\ (x e. B /\ ph)))
7	6	exbii 1582	. . 3 |- (E.x(x e. B /\ (A e. x /\ ph)) <-> E.x(A e. x /\ (x e. B /\ ph)))
8	5, 7	bitri 240	. 2 |- (E.x e. B (A e. x /\ ph) <-> E.x(A e. x /\ (x e. B /\ ph)))
9	1, 4, 8	3bitr4i 268	1 |- (A e. U.{x e. B | ph} <-> E.x e. B (A e. x /\ ph))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   e. wcel 1710  {cab 2339  E.wrex 2615  {crab 2618  U.cuni 3891

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-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-rab 2623  df-v 2861  df-uni 3892

This theorem is referenced by: (None)