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Theorem reximdva0 3632
Description: Restricted existence deduced from non-empty 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 3630 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 reximdva0.1 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ps )
32ex 424 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  ps ) )
43ancld 537 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  ( x  e.  A  /\  ps ) ) )
54eximdv 1632 . . . 4  |-  ( ph  ->  ( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ps ) ) )
65imp 419 . . 3  |-  ( (
ph  /\  E. x  x  e.  A )  ->  E. x ( x  e.  A  /\  ps ) )
71, 6sylan2b 462 . 2  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. x
( x  e.  A  /\  ps ) )
8 df-rex 2704 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
97, 8sylibr 204 1  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    e. wcel 1725    =/= wne 2599   E.wrex 2699   (/)c0 3621
This theorem is referenced by:  hashgt12el  11675  cstucnd  18307  supxrnemnf  24120  kerunit  24254  usgfiregdegfi  28315  elpaddn0  30535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2704  df-v 2951  df-dif 3316  df-nul 3622
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