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Theorem abexex 6756
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 2810 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 abexex.2 . . . . . 6  |-  ( ph  ->  x  e.  A )
32pm4.71ri 631 . . . . 5  |-  ( ph  <->  ( x  e.  A  /\  ph ) )
43exbii 1672 . . . 4  |-  ( E. x ph  <->  E. x
( x  e.  A  /\  ph ) )
51, 4bitr4i 252 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ph )
65abbii 2588 . 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 6754 . 2  |-  { y  |  E. x  e.  A  ph }  e.  _V
106, 9eqeltrri 2539 1  |-  { y  |  E. x ph }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   E.wex 1617    e. wcel 1823   {cab 2439   E.wrex 2805   _Vcvv 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578
This theorem is referenced by:  brdom7disj  8900  brdom6disj  8901
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