Theorem elunirab 2510

Description: Membership in union of a class abstraction.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem elunirab

Step	Hyp	Ref	Expression
1		eluniab 2509	. 2 |- (A e. U.{x | (x e. B /\ ph)} <-> E.x(A e. x /\ (x e. B /\ ph)))
2		df-rab 1650	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
3	2	unieqi 2507	. . 3 |- U.{x e. B | ph} = U.{x | (x e. B /\ ph)}
4	3	eleq2i 1536	. 2 |- (A e. U.{x e. B | ph} <-> A e. U.{x | (x e. B /\ ph)})
5		df-rex 1648	. . 3 |- (E.x e. B (A e. x /\ ph) <-> E.x(x e. B /\ (A e. x /\ ph)))
6		an12 484	. . . 4 |- ((x e. B /\ (A e. x /\ ph)) <-> (A e. x /\ (x e. B /\ ph)))
7	6	exbii 1050	. . 3 |- (E.x(x e. B /\ (A e. x /\ ph)) <-> E.x(A e. x /\ (x e. B /\ ph)))
8	5, 7	bitr 173	. 2 |- (E.x e. B (A e. x /\ ph) <-> E.x(A e. x /\ (x e. B /\ ph)))
9	1, 4, 8	3bitr4 183	1 |- (A e. U.{x e. B | ph} <-> E.x e. B (A e. x /\ ph))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 957  E.wex 979  {cab 1462  E.wrex 1644  {crab 1646  U.cuni 2499

This theorem is referenced by:  clsval2 7645  ntrss2 7650  ntreq0 7668  cncnplem4 7737

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-rex 1648  df-rab 1650  df-v 1809  df-uni 2500

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