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Theorem reximdva0 3466
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 3464 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 reximdva0.1 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ps )
32ex 423 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  ps ) )
43ancld 536 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  ( x  e.  A  /\  ps ) ) )
54eximdv 1608 . . . 4  |-  ( ph  ->  ( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ps ) ) )
65imp 418 . . 3  |-  ( (
ph  /\  E. x  x  e.  A )  ->  E. x ( x  e.  A  /\  ps ) )
71, 6sylan2b 461 . 2  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. x
( x  e.  A  /\  ps ) )
8 df-rex 2549 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
97, 8sylibr 203 1  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    e. wcel 1684    =/= wne 2446   E.wrex 2544   (/)c0 3455
This theorem is referenced by:  supxrnemnf  23256  intopcoaconlem3b  25538  elpaddn0  29989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-v 2790  df-dif 3155  df-nul 3456
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