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Theorem abexex 5879
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 2365 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 abexex.2 . . . . . 6  |-  ( ph  ->  x  e.  A )
32pm4.71ri 384 . . . . 5  |-  ( ph  <->  ( x  e.  A  /\  ph ) )
43exbii 1541 . . . 4  |-  ( E. x ph  <->  E. x
( x  e.  A  /\  ph ) )
51, 4bitr4i 185 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ph )
65abbii 2203 . 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 5877 . 2  |-  { y  |  E. x  e.  A  ph }  e.  _V
106, 9eqeltrri 2161 1  |-  { y  |  E. x ph }  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   E.wex 1426    e. wcel 1438   {cab 2074   E.wrex 2360   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010
This theorem is referenced by: (None)
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