Theorem eluniab 3808

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 3799	. 2 |- ( A e. U. { x | ph } <-> E. y ( A e. y /\ y e. { x | ph } ) )
2		nfv 1521	. . . 4 |- F/ x A e. y
3		nfsab1 2160	. . . 4 |- F/ x y e. { x | ph }
4	2, 3	nfan 1558	. . 3 |- F/ x ( A e. y /\ y e. { x | ph } )
5		nfv 1521	. . 3 |- F/ y ( A e. x /\ ph )
6		eleq2 2234	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
7		eleq1 2233	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2158	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	bitrdi 195	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10	6, 9	anbi12d 470	. . 3 |- ( y = x -> ( ( A e. y /\ y e. { x | ph } ) <-> ( A e. x /\ ph ) ) )
11	4, 5, 10	cbvex 1749	. 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 1485   e. wcel 2141   {cab 2156   U.cuni 3796

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-uni 3797

This theorem is referenced by:  elunirab  3809  dfiun2g  3905  inuni  4141  snnex  4433  eliota  5186  elfv  5494  unielxp  6153  tfrlem9  6298  tfr0dm  6301  metrest  13300