Theorem eluniab 3876

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 3867	. 2 |- ( A e. U. { x | ph } <-> E. y ( A e. y /\ y e. { x | ph } ) )
2		nfv 1552	. . . 4 |- F/ x A e. y
3		nfsab1 2197	. . . 4 |- F/ x y e. { x | ph }
4	2, 3	nfan 1589	. . 3 |- F/ x ( A e. y /\ y e. { x | ph } )
5		nfv 1552	. . 3 |- F/ y ( A e. x /\ ph )
6		eleq2 2271	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
7		eleq1 2270	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2195	. . . . 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 1780	. 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 1516   e. wcel 2178   {cab 2193   U.cuni 3864

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-uni 3865

This theorem is referenced by:  elunirab  3877  dfiun2g  3973  inuni  4215  snnex  4513  eliota  5278  elfv  5597  unielxp  6283  tfrlem9  6428  tfr0dm  6431  metrest  15093