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Theorem eluniab 3839
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 3830 . 2  |-  ( A  e.  U. { x  |  ph }  <->  E. y
( A  e.  y  /\  y  e.  {
x  |  ph }
) )
2 nfv 1605 . . . 4  |-  F/ x  A  e.  y
3 nfsab1 2273 . . . 4  |-  F/ x  y  e.  { x  |  ph }
42, 3nfan 1771 . . 3  |-  F/ x
( A  e.  y  /\  y  e.  {
x  |  ph }
)
5 nfv 1605 . . 3  |-  F/ y ( A  e.  x  /\  ph )
6 eleq2 2344 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
7 eleq1 2343 . . . . 5  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
8 abid 2271 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
97, 8syl6bb 252 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
106, 9anbi12d 691 . . 3  |-  ( y  =  x  ->  (
( A  e.  y  /\  y  e.  {
x  |  ph }
)  <->  ( A  e.  x  /\  ph )
) )
114, 5, 10cbvex 1925 . 2  |-  ( E. y ( A  e.  y  /\  y  e. 
{ x  |  ph } )  <->  E. x
( A  e.  x  /\  ph ) )
121, 11bitri 240 1  |-  ( A  e.  U. { x  |  ph }  <->  E. x
( A  e.  x  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   U.cuni 3827
This theorem is referenced by:  elunirab  3840  dfiun2g  3935  inuni  4173  snnex  4524  elfv  5523  unielxp  6158  tfrlem9  6401  dfac5lem2  7751  fin23lem30  7968  metrest  18070  aannenlem2  19709  wfrlem12  24267  frrlem11  24293  dfiota3  24462
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-uni 3828
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