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Theorem elunirab 3988
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 3987 . 2  |-  ( A  e.  U. { x  |  ( x  e.  B  /\  ph ) } 
<->  E. x ( A  e.  x  /\  (
x  e.  B  /\  ph ) ) )
2 df-rab 2675 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
32unieqi 3985 . . 3  |-  U. {
x  e.  B  |  ph }  =  U. {
x  |  ( x  e.  B  /\  ph ) }
43eleq2i 2468 . 2  |-  ( A  e.  U. { x  e.  B  |  ph }  <->  A  e.  U. { x  |  ( x  e.  B  /\  ph ) } )
5 df-rex 2672 . . 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 1589 . . 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 1547    e. wcel 1721   {cab 2390   E.wrex 2667   {crab 2670   U.cuni 3975
This theorem is referenced by:  neiptopuni  17149  cmpcov2  17407  tgcmp  17418  hauscmplem  17423  concompid  17447  alexsubALT  18035  cvmliftlem15  24938  fnessref  26263  cover2  26305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-uni 3976
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