Theorem eluniab 3899

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 3890	. 2 |- ( A e. U. { x | ph } <-> E. y ( A e. y /\ y e. { x | ph } ) )
2		nfv 1574	. . . 4 |- F/ x A e. y
3		nfsab1 2219	. . . 4 |- F/ x y e. { x | ph }
4	2, 3	nfan 1611	. . 3 |- F/ x ( A e. y /\ y e. { x | ph } )
5		nfv 1574	. . 3 |- F/ y ( A e. x /\ ph )
6		eleq2 2293	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
7		eleq1 2292	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2217	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	bitrdi 196	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10	6, 9	anbi12d 473	. . 3 |- ( y = x -> ( ( A e. y /\ y e. { x | ph } ) <-> ( A e. x /\ ph ) ) )
11	4, 5, 10	cbvex 1802	. 2 |- ( E. y ( A e. y /\ y e. { x | ph } ) <-> E. x ( A e. x /\ ph ) )
12	1, 11	bitri 184	1 |- ( A e. U. { x | ph } <-> E. x ( A e. x /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1538   e. wcel 2200   {cab 2215   U.cuni 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-uni 3888

This theorem is referenced by:  elunirab  3900  dfiun2g  3996  inuni  4238  snnex  4538  eliota  5305  elfv  5624  unielxp  6318  tfrlem9  6463  tfr0dm  6466  metrest  15174