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Theorem elunirab 4028
Description: Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
Assertion
Ref Expression
elunirab  |-  ( A  e.  U. { x  e.  B  |  ph }  <->  E. x  e.  B  ( A  e.  x  /\  ph ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem elunirab
StepHypRef Expression
1 eluniab 4027 . 2  |-  ( A  e.  U. { x  |  ( x  e.  B  /\  ph ) } 
<->  E. x ( A  e.  x  /\  (
x  e.  B  /\  ph ) ) )
2 df-rab 2714 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
32unieqi 4025 . . 3  |-  U. {
x  e.  B  |  ph }  =  U. {
x  |  ( x  e.  B  /\  ph ) }
43eleq2i 2500 . 2  |-  ( A  e.  U. { x  e.  B  |  ph }  <->  A  e.  U. { x  |  ( x  e.  B  /\  ph ) } )
5 df-rex 2711 . . 3  |-  ( E. x  e.  B  ( A  e.  x  /\  ph )  <->  E. x ( x  e.  B  /\  ( A  e.  x  /\  ph ) ) )
6 an12 773 . . . 4  |-  ( ( x  e.  B  /\  ( A  e.  x  /\  ph ) )  <->  ( A  e.  x  /\  (
x  e.  B  /\  ph ) ) )
76exbii 1592 . . 3  |-  ( E. x ( x  e.  B  /\  ( A  e.  x  /\  ph ) )  <->  E. x
( A  e.  x  /\  ( x  e.  B  /\  ph ) ) )
85, 7bitri 241 . 2  |-  ( E. x  e.  B  ( A  e.  x  /\  ph )  <->  E. x ( A  e.  x  /\  (
x  e.  B  /\  ph ) ) )
91, 4, 83bitr4i 269 1  |-  ( A  e.  U. { x  e.  B  |  ph }  <->  E. x  e.  B  ( A  e.  x  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    e. wcel 1725   {cab 2422   E.wrex 2706   {crab 2709   U.cuni 4015
This theorem is referenced by:  neiptopuni  17194  cmpcov2  17453  tgcmp  17464  hauscmplem  17469  concompid  17494  alexsubALT  18082  cvmliftlem15  24985  fnessref  26373  cover2  26415
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-uni 4016
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