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Theorem reximdva0 3575
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 3573 . . 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 1629 . . . 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 2648 . 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 1547    e. wcel 1717    =/= wne 2543   E.wrex 2643   (/)c0 3564
This theorem is referenced by:  hashgt12el  11602  cstucnd  18228  supxrnemnf  23956  kerunit  24070  elpaddn0  29965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-rex 2648  df-v 2894  df-dif 3259  df-nul 3565
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