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Theorem elunirab 3840
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 3839 . 2  |-  ( A  e.  U. { x  |  ( x  e.  B  /\  ph ) } 
<->  E. x ( A  e.  x  /\  (
x  e.  B  /\  ph ) ) )
2 df-rab 2552 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
32unieqi 3837 . . 3  |-  U. {
x  e.  B  |  ph }  =  U. {
x  |  ( x  e.  B  /\  ph ) }
43eleq2i 2347 . 2  |-  ( A  e.  U. { x  e.  B  |  ph }  <->  A  e.  U. { x  |  ( x  e.  B  /\  ph ) } )
5 df-rex 2549 . . 3  |-  ( E. x  e.  B  ( A  e.  x  /\  ph )  <->  E. x ( x  e.  B  /\  ( A  e.  x  /\  ph ) ) )
6 an12 772 . . . 4  |-  ( ( x  e.  B  /\  ( A  e.  x  /\  ph ) )  <->  ( A  e.  x  /\  (
x  e.  B  /\  ph ) ) )
76exbii 1569 . . 3  |-  ( E. x ( x  e.  B  /\  ( A  e.  x  /\  ph ) )  <->  E. x
( A  e.  x  /\  ( x  e.  B  /\  ph ) ) )
85, 7bitri 240 . 2  |-  ( E. x  e.  B  ( A  e.  x  /\  ph )  <->  E. x ( A  e.  x  /\  (
x  e.  B  /\  ph ) ) )
91, 4, 83bitr4i 268 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 176    /\ wa 358   E.wex 1528    e. wcel 1684   {cab 2269   E.wrex 2544   {crab 2547   U.cuni 3827
This theorem is referenced by:  cmpcov2  17117  tgcmp  17128  hauscmplem  17133  concompid  17157  alexsubALT  17745  cvmliftlem15  23240  fnessref  25705  cover2  25770
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-rex 2549  df-rab 2552  df-v 2790  df-uni 3828
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