Theorem eluniab 3942

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 3933	. 2 |- ( A e. U. { x | ph } <-> E. y ( A e. y /\ y e. { x | ph } ) )
2		nfv 1581	. . . 4 |- F/ x A e. y
3		nfsab1 2228	. . . 4 |- F/ x y e. { x | ph }
4	2, 3	nfan 1618	. . 3 |- F/ x ( A e. y /\ y e. { x | ph } )
5		nfv 1581	. . 3 |- F/ y ( A e. x /\ ph )
6		eleq2 2302	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
7		eleq1 2301	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2226	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	bitrdi 196	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10	6, 9	anbi12d 477	. . 3 |- ( y = x -> ( ( A e. y /\ y e. { x | ph } ) <-> ( A e. x /\ ph ) ) )
11	4, 5, 10	cbvex 1809	. 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 1545   e. wcel 2209   {cab 2224   U.cuni 3930

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-uni 3931

This theorem is referenced by:  elunirab  3943  dfiun2g  4039  inuni  4286  snnex  4589  eliota  5360  elfv  5688  unielxp  6398  tfrlem9  6580  tfr0dm  6583  metrest  15530