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Theorem abexex 6559
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 2719 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 abexex.2 . . . . . 6  |-  ( ph  ->  x  e.  A )
32pm4.71ri 628 . . . . 5  |-  ( ph  <->  ( x  e.  A  /\  ph ) )
43exbii 1639 . . . 4  |-  ( E. x ph  <->  E. x
( x  e.  A  /\  ph ) )
51, 4bitr4i 252 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ph )
65abbii 2553 . 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 6557 . 2  |-  { y  |  E. x  e.  A  ph }  e.  _V
106, 9eqeltrri 2512 1  |-  { y  |  E. x ph }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1591    e. wcel 1761   {cab 2427   E.wrex 2714   _Vcvv 2970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423
This theorem is referenced by:  brdom7disj  8694  brdom6disj  8695
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