Theorem eluniab 3592

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 3583	. 2 |- ( A e. U. { x | ph } <-> E. y ( A e. y /\ y e. { x | ph } ) )
2		nfv 1421	. . . 4 |- F/ x A e. y
3		nfsab1 2030	. . . 4 |- F/ x y e. { x | ph }
4	2, 3	nfan 1457	. . 3 |- F/ x ( A e. y /\ y e. { x | ph } )
5		nfv 1421	. . 3 |- F/ y ( A e. x /\ ph )
6		eleq2 2101	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
7		eleq1 2100	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2028	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	syl6bb 185	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10	6, 9	anbi12d 442	. . 3 |- ( y = x -> ( ( A e. y /\ y e. { x | ph } ) <-> ( A e. x /\ ph ) ) )
11	4, 5, 10	cbvex 1639	. 2 |- ( E. y ( A e. y /\ y e. { x | ph } ) <-> E. x ( A e. x /\ ph ) )
12	1, 11	bitri 173	1 |- ( A e. U. { x | ph } <-> E. x ( A e. x /\ ph ) )

Syntax hints:   /\ wa 97   <-> wb 98   E.wex 1381   e. wcel 1393   {cab 2026   U.cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-uni 3581

This theorem is referenced by:  elunirab  3593  dfiun2g  3689  inuni  3909  snnex  4181  elfv  5176  unielxp  5800  tfrlem9  5935  tfr0  5937