Theorem eluniab 3756

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 3747	. 2 |- ( A e. U. { x | ph } <-> E. y ( A e. y /\ y e. { x | ph } ) )
2		nfv 1509	. . . 4 |- F/ x A e. y
3		nfsab1 2130	. . . 4 |- F/ x y e. { x | ph }
4	2, 3	nfan 1545	. . 3 |- F/ x ( A e. y /\ y e. { x | ph } )
5		nfv 1509	. . 3 |- F/ y ( A e. x /\ ph )
6		eleq2 2204	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
7		eleq1 2203	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2128	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	syl6bb 195	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10	6, 9	anbi12d 465	. . 3 |- ( y = x -> ( ( A e. y /\ y e. { x | ph } ) <-> ( A e. x /\ ph ) ) )
11	4, 5, 10	cbvex 1730	. 2 |- ( E. y ( A e. y /\ y e. { x | ph } ) <-> E. x ( A e. x /\ ph ) )
12	1, 11	bitri 183	1 |- ( A e. U. { x | ph } <-> E. x ( A e. x /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1469   e. wcel 1481   {cab 2126   U.cuni 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-uni 3745

This theorem is referenced by:  elunirab  3757  dfiun2g  3853  inuni  4088  snnex  4377  elfv  5427  unielxp  6080  tfrlem9  6224  tfr0dm  6227  metrest  12714