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Theorem eluniab 3855
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 3846 . 2  |-  ( A  e.  U. { x  |  ph }  <->  E. y
( A  e.  y  /\  y  e.  {
x  |  ph }
) )
2 nfv 1609 . . . 4  |-  F/ x  A  e.  y
3 nfsab1 2286 . . . 4  |-  F/ x  y  e.  { x  |  ph }
42, 3nfan 1783 . . 3  |-  F/ x
( A  e.  y  /\  y  e.  {
x  |  ph }
)
5 nfv 1609 . . 3  |-  F/ y ( A  e.  x  /\  ph )
6 eleq2 2357 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
7 eleq1 2356 . . . . 5  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
8 abid 2284 . . . . 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 1938 . 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   U.cuni 3843
This theorem is referenced by:  elunirab  3856  dfiun2g  3951  inuni  4189  snnex  4540  elfv  5539  unielxp  6174  tfrlem9  6417  dfac5lem2  7767  fin23lem30  7984  metrest  18086  aannenlem2  19725  wfrlem12  24338  frrlem11  24364  dfiota3  24533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-uni 3844
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