Theorem eluniab 2513

Description: Membership in union of a class abstraction.

Assertion

Ref	Expression
eluniab	|- (A e. U.{x | ph} <-> E.x(A e. x /\ ph))

Distinct variable group:   x,A

Proof of Theorem eluniab

Step	Hyp	Ref	Expression
1		eluni 2506	. 2 |- (A e. U.{x | ph} <-> E.y(A e. y /\ y e. {x | ph}))
2		ax-17 971	. . . 4 |- (A e. y -> A.x A e. y)
3		hbab1 1466	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
4	2, 3	hban 1009	. . 3 |- ((A e. y /\ y e. {x | ph}) -> A.x(A e. y /\ y e. {x | ph}))
5		ax-17 971	. . 3 |- ((A e. x /\ ph) -> A.y(A e. x /\ ph))
6		eleq2 1535	. . . . 5 |- (y = x -> (A e. y <-> A e. x))
7		eleq1 1534	. . . . 5 |- (y = x -> (y e. {x | ph} <-> x e. {x | ph}))
8	6, 7	anbi12d 628	. . . 4 |- (y = x -> ((A e. y /\ y e. {x | ph}) <-> (A e. x /\ x e. {x | ph})))
9		abid 1465	. . . . 5 |- (x e. {x | ph} <-> ph)
10	9	anbi2i 480	. . . 4 |- ((A e. x /\ x e. {x | ph}) <-> (A e. x /\ ph))
11	8, 10	syl6bb 536	. . 3 |- (y = x -> ((A e. y /\ y e. {x | ph}) <-> (A e. x /\ ph)))
12	4, 5, 11	cbvex 1166	. 2 |- (E.y(A e. y /\ y e. {x | ph}) <-> E.x(A e. x /\ ph))
13	1, 12	bitr 173	1 |- (A e. U.{x | ph} <-> E.x(A e. x /\ ph))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  U.cuni 2503

This theorem is referenced by:  elunirab 2514  elfv 3722  funiunfv 3866  tfrlem9 3919  unielxp 4107  aceq5lem2 4736  tgval3t 7625  subbas2OLD 7645  ntunte 10439  rcfpfillem3 10589  rcfpfillem3OLD 10590

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-uni 2504

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