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Theorem eluniab 4019
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 4010 . 2  |-  ( A  e.  U. { x  |  ph }  <->  E. y
( A  e.  y  /\  y  e.  {
x  |  ph }
) )
2 nfv 1629 . . . 4  |-  F/ x  A  e.  y
3 nfsab1 2425 . . . 4  |-  F/ x  y  e.  { x  |  ph }
42, 3nfan 1846 . . 3  |-  F/ x
( A  e.  y  /\  y  e.  {
x  |  ph }
)
5 nfv 1629 . . 3  |-  F/ y ( A  e.  x  /\  ph )
6 eleq2 2496 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
7 eleq1 2495 . . . . 5  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
8 abid 2423 . . . . 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 1983 . 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 1550    e. wcel 1725   {cab 2421   U.cuni 4007
This theorem is referenced by:  elunirab  4020  dfiun2g  4115  inuni  4354  snnex  4705  elfv  5718  unielxp  6377  tfrlem9  6638  dfac5lem2  7995  fin23lem30  8212  metrest  18544  aannenlem2  20236  wfrlem12  25534  frrlem11  25559  dfiota3  25733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-uni 4008
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