Theorem eluniab 3801

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 3792	. 2 |- ( A e. U. { x | ph } <-> E. y ( A e. y /\ y e. { x | ph } ) )
2		nfv 1516	. . . 4 |- F/ x A e. y
3		nfsab1 2155	. . . 4 |- F/ x y e. { x | ph }
4	2, 3	nfan 1553	. . 3 |- F/ x ( A e. y /\ y e. { x | ph } )
5		nfv 1516	. . 3 |- F/ y ( A e. x /\ ph )
6		eleq2 2230	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
7		eleq1 2229	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2153	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	bitrdi 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 1744	. 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 1480   e. wcel 2136   {cab 2151   U.cuni 3789

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-uni 3790

This theorem is referenced by:  elunirab  3802  dfiun2g  3898  inuni  4134  snnex  4426  elfv  5484  unielxp  6142  tfrlem9  6287  tfr0dm  6290  metrest  13146