Theorem eluniab 3743

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 3734	. 2 |- ( A e. U. { x | ph } <-> E. y ( A e. y /\ y e. { x | ph } ) )
2		nfv 1508	. . . 4 |- F/ x A e. y
3		nfsab1 2127	. . . 4 |- F/ x y e. { x | ph }
4	2, 3	nfan 1544	. . 3 |- F/ x ( A e. y /\ y e. { x | ph } )
5		nfv 1508	. . 3 |- F/ y ( A e. x /\ ph )
6		eleq2 2201	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
7		eleq1 2200	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2125	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	syl6bb 195	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10	6, 9	anbi12d 464	. . 3 |- ( y = x -> ( ( A e. y /\ y e. { x | ph } ) <-> ( A e. x /\ ph ) ) )
11	4, 5, 10	cbvex 1729	. 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 1468   e. wcel 1480   {cab 2123   U.cuni 3731

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-uni 3732

This theorem is referenced by:  elunirab  3744  dfiun2g  3840  inuni  4075  snnex  4364  elfv  5412  unielxp  6065  tfrlem9  6209  tfr0dm  6212  metrest  12664