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Theorem abexex 5992
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 2399 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 abexex.2 . . . . . 6  |-  ( ph  ->  x  e.  A )
32pm4.71ri 389 . . . . 5  |-  ( ph  <->  ( x  e.  A  /\  ph ) )
43exbii 1569 . . . 4  |-  ( E. x ph  <->  E. x
( x  e.  A  /\  ph ) )
51, 4bitr4i 186 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ph )
65abbii 2233 . 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 5990 . 2  |-  { y  |  E. x  e.  A  ph }  e.  _V
106, 9eqeltrri 2191 1  |-  { y  |  E. x ph }  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1453    e. wcel 1465   {cab 2103   E.wrex 2394   _Vcvv 2660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101
This theorem is referenced by: (None)
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