Theorem eluniab 3613

Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem eluniab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 3604	. 2 |- ( A e. U. { x | ph } <-> E. y ( A e. y /\ y e. { x | ph } ) )
2		nfv 1461	. . . 4 |- F/ x A e. y
3		nfsab1 2071	. . . 4 |- F/ x y e. { x | ph }
4	2, 3	nfan 1497	. . 3 |- F/ x ( A e. y /\ y e. { x | ph } )
5		nfv 1461	. . 3 |- F/ y ( A e. x /\ ph )
6		eleq2 2142	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
7		eleq1 2141	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2069	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	syl6bb 194	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10	6, 9	anbi12d 456	. . 3 |- ( y = x -> ( ( A e. y /\ y e. { x | ph } ) <-> ( A e. x /\ ph ) ) )
11	4, 5, 10	cbvex 1679	. 2 |- ( E. y ( A e. y /\ y e. { x | ph } ) <-> E. x ( A e. x /\ ph ) )
12	1, 11	bitri 182	1 |- ( A e. U. { x | ph } <-> E. x ( A e. x /\ ph ) )

Syntax hints:   /\ wa 102   <-> wb 103   E.wex 1421   e. wcel 1433   {cab 2067   U.cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-uni 3602

This theorem is referenced by:  elunirab  3614  dfiun2g  3710  inuni  3930  snnex  4199  elfv  5196  unielxp  5820  tfrlem9  5958  tfr0  5960