Theorem eluniab 3903

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 3894	. 2 |- (A e. U.{x | ph} <-> E.y(A e. y /\ y e. {x | ph}))
2		nfv 1619	. . . 4 |- F/x A e. y
3		nfsab1 2343	. . . 4 |- F/x y e. {x | ph}
4	2, 3	nfan 1824	. . 3 |- F/x(A e. y /\ y e. {x | ph})
5		nfv 1619	. . 3 |- F/y(A e. x /\ ph)
6		eleq2 2414	. . . 4 |- (y = x -> (A e. y <-> A e. x))
7		eleq1 2413	. . . . 5 |- (y = x -> (y e. {x | ph} <-> x e. {x | ph}))
8		abid 2341	. . . . 5 |- (x e. {x | ph} <-> ph)
9	7, 8	syl6bb 252	. . . 4 |- (y = x -> (y e. {x | ph} <-> ph))
10	6, 9	anbi12d 691	. . 3 |- (y = x -> ((A e. y /\ y e. {x | ph}) <-> (A e. x /\ ph)))
11	4, 5, 10	cbvex 1985	. 2 |- (E.y(A e. y /\ y e. {x | ph}) <-> E.x(A e. x /\ ph))
12	1, 11	bitri 240	1 |- (A e. U.{x | ph} <-> E.x(A e. x /\ ph))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  U.cuni 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-uni 3892

This theorem is referenced by:  elunirab  3904  dfiun2g  3999  eqtfinrelk  4486  elfv  5326  funiunfv  5467  tcfnex  6244