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Theorem abexex 6005
Description: A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
Hypotheses
Ref Expression
abexex.1  |-  A  e. 
_V
abexex.2  |-  ( ph  ->  x  e.  A )
abexex.3  |-  { y  |  ph }  e.  _V
Assertion
Ref Expression
abexex  |-  { y  |  E. x ph }  e.  _V
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem abexex
StepHypRef Expression
1 df-rex 2713 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 abexex.2 . . . . . 6  |-  ( ph  ->  x  e.  A )
32pm4.71ri 616 . . . . 5  |-  ( ph  <->  ( x  e.  A  /\  ph ) )
43exbii 1593 . . . 4  |-  ( E. x ph  <->  E. x
( x  e.  A  /\  ph ) )
51, 4bitr4i 245 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ph )
65abbii 2550 . 2  |-  { y  |  E. x  e.  A  ph }  =  { y  |  E. x ph }
7 abexex.1 . . 3  |-  A  e. 
_V
8 abexex.3 . . 3  |-  { y  |  ph }  e.  _V
97, 8abrexex2 6003 . 2  |-  { y  |  E. x  e.  A  ph }  e.  _V
106, 9eqeltrri 2509 1  |-  { y  |  E. x ph }  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    e. wcel 1726   {cab 2424   E.wrex 2708   _Vcvv 2958
This theorem is referenced by:  brdom7disj  8411  brdom6disj  8412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464
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