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Theorem eluniab 3962
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.
StepHypRef Expression
1 eluni 3953 . 2  |-  ( A  e.  U. { x  |  ph }  <->  E. y
( A  e.  y  /\  y  e.  {
x  |  ph }
) )
2 nfv 1626 . . . 4  |-  F/ x  A  e.  y
3 nfsab1 2370 . . . 4  |-  F/ x  y  e.  { x  |  ph }
42, 3nfan 1836 . . 3  |-  F/ x
( A  e.  y  /\  y  e.  {
x  |  ph }
)
5 nfv 1626 . . 3  |-  F/ y ( A  e.  x  /\  ph )
6 eleq2 2441 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
7 eleq1 2440 . . . . 5  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
8 abid 2368 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
97, 8syl6bb 253 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
106, 9anbi12d 692 . . 3  |-  ( y  =  x  ->  (
( A  e.  y  /\  y  e.  {
x  |  ph }
)  <->  ( A  e.  x  /\  ph )
) )
114, 5, 10cbvex 2011 . 2  |-  ( E. y ( A  e.  y  /\  y  e. 
{ x  |  ph } )  <->  E. x
( A  e.  x  /\  ph ) )
121, 11bitri 241 1  |-  ( A  e.  U. { x  |  ph }  <->  E. x
( A  e.  x  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    e. wcel 1717   {cab 2366   U.cuni 3950
This theorem is referenced by:  elunirab  3963  dfiun2g  4058  inuni  4296  snnex  4646  elfv  5659  unielxp  6317  tfrlem9  6575  dfac5lem2  7931  fin23lem30  8148  metrest  18437  aannenlem2  20106  wfrlem12  25284  frrlem11  25310  dfiota3  25479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-uni 3951
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