Theorem elunirab 3190

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 3189	. 2 |- (A e. U.{x | (x e. B /\ ph)} <-> E.x(A e. x /\ (x e. B /\ ph)))
2		df-rab 2112	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
3	2	unieqi 3187	. . 3 |- U.{x e. B | ph} = U.{x | (x e. B /\ ph)}
4	3	eleq2i 1961	. 2 |- (A e. U.{x e. B | ph} <-> A e. U.{x | (x e. B /\ ph)})
5		df-rex 2110	. . 3 |- (E.x e. B (A e. x /\ ph) <-> E.x(x e. B /\ (A e. x /\ ph)))
6		an12 542	. . . 4 |- ((x e. B /\ (A e. x /\ ph)) <-> (A e. x /\ (x e. B /\ ph)))
7	6	exbii 1398	. . 3 |- (E.x(x e. B /\ (A e. x /\ ph)) <-> E.x(A e. x /\ (x e. B /\ ph)))
8	5, 7	bitri 190	. 2 |- (E.x e. B (A e. x /\ ph) <-> E.x(A e. x /\ (x e. B /\ ph)))
9	1, 4, 8	3bitr4i 200	1 |- (A e. U.{x e. B | ph} <-> E.x e. B (A e. x /\ ph))

Syntax hints:   <-> wb 163   /\ wa 240   e. wcel 1300  E.wex 1326  {cab 1871  E.wrex 2106  {crab 2108  U.cuni 3177

This theorem is referenced by:  clsval2 8961  ntrss2 8966  ntreq0 8984  cncnplem4 9054  hscptsscld 15434  alexsub 15441  fnessref 15503  locfincomp 15514  cover2 15673

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-rab 2112  df-v 2294  df-uni 3178

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