Theorem eluniab 2509

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 2502	. 2 |- (A e. U.{x | ph} <-> E.y(A e. y /\ y e. {x | ph}))
2		ax-17 970	. . . 4 |- (A e. y -> A.x A e. y)
3		hbab1 1465	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
4	2, 3	hban 1008	. . 3 |- ((A e. y /\ y e. {x | ph}) -> A.x(A e. y /\ y e. {x | ph}))
5		ax-17 970	. . 3 |- ((A e. x /\ ph) -> A.y(A e. x /\ ph))
6		eleq2 1533	. . . . 5 |- (y = x -> (A e. y <-> A e. x))
7		eleq1 1532	. . . . 5 |- (y = x -> (y e. {x | ph} <-> x e. {x | ph}))
8	6, 7	anbi12d 627	. . . 4 |- (y = x -> ((A e. y /\ y e. {x | ph}) <-> (A e. x /\ x e. {x | ph})))
9		abid 1464	. . . . 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 535	. . 3 |- (y = x -> ((A e. y /\ y e. {x | ph}) <-> (A e. x /\ ph)))
12	4, 5, 11	cbvex 1165	. 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 955   e. wcel 957  E.wex 979  {cab 1462  U.cuni 2499

This theorem is referenced by:  elunirab 2510  elfv 3717  funiunfv 3861  tfrlem9 3914  unielxp 4100  aceq5lem2 4719  tgval3t 7585  subbas2 7605  ntunte 10398  rcfpfillem3 10513

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-uni 2500

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