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Theorem eluniab 3633
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 3624 . 2  |-  ( A  e.  U. { x  |  ph }  <->  E. y
( A  e.  y  /\  y  e.  {
x  |  ph }
) )
2 nfv 1462 . . . 4  |-  F/ x  A  e.  y
3 nfsab1 2073 . . . 4  |-  F/ x  y  e.  { x  |  ph }
42, 3nfan 1498 . . 3  |-  F/ x
( A  e.  y  /\  y  e.  {
x  |  ph }
)
5 nfv 1462 . . 3  |-  F/ y ( A  e.  x  /\  ph )
6 eleq2 2146 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
7 eleq1 2145 . . . . 5  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
8 abid 2071 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
97, 8syl6bb 194 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
106, 9anbi12d 457 . . 3  |-  ( y  =  x  ->  (
( A  e.  y  /\  y  e.  {
x  |  ph }
)  <->  ( A  e.  x  /\  ph )
) )
114, 5, 10cbvex 1681 . 2  |-  ( E. y ( A  e.  y  /\  y  e. 
{ x  |  ph } )  <->  E. x
( A  e.  x  /\  ph ) )
121, 11bitri 182 1  |-  ( A  e.  U. { x  |  ph }  <->  E. x
( A  e.  x  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   E.wex 1422    e. wcel 1434   {cab 2069   U.cuni 3621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-uni 3622
This theorem is referenced by:  elunirab  3634  dfiun2g  3730  inuni  3950  snnex  4227  elfv  5228  unielxp  5852  tfrlem9  5989  tfr0dm  5992
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