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Theorem reximdva0 3795
Description: Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)
Hypothesis
Ref Expression
reximdva0.1  |-  ( (
ph  /\  x  e.  A )  ->  ps )
Assertion
Ref Expression
reximdva0  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. x  e.  A  ps )
Distinct variable groups:    x, A    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem reximdva0
StepHypRef Expression
1 n0 3793 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 reximdva0.1 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ps )
32ex 432 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  ps ) )
43ancld 551 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  ( x  e.  A  /\  ps ) ) )
54eximdv 1715 . . . 4  |-  ( ph  ->  ( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ps ) ) )
65imp 427 . . 3  |-  ( (
ph  /\  E. x  x  e.  A )  ->  E. x ( x  e.  A  /\  ps ) )
71, 6sylan2b 473 . 2  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. x
( x  e.  A  /\  ps ) )
8 df-rex 2810 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
97, 8sylibr 212 1  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. x  e.  A  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   E.wex 1617    e. wcel 1823    =/= wne 2649   E.wrex 2805   (/)c0 3783
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-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rex 2810  df-v 3108  df-dif 3464  df-nul 3784
This theorem is referenced by:  hashgt12el  12468  refun0  20185  cstucnd  20956  supxrnemnf  27820  kerunit  28051  elpaddn0  35940
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