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Theorem rexex 2503
Description: Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
Assertion
Ref Expression
rexex  |-  ( E. x  e.  A  ph  ->  E. x ph )

Proof of Theorem rexex
StepHypRef Expression
1 df-rex 2441 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 simpr 109 . . 3  |-  ( ( x  e.  A  /\  ph )  ->  ph )
32eximi 1580 . 2  |-  ( E. x ( x  e.  A  /\  ph )  ->  E. x ph )
41, 3sylbi 120 1  |-  ( E. x  e.  A  ph  ->  E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1472    e. wcel 2128   E.wrex 2436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-rex 2441
This theorem is referenced by:  reu3  2902  rmo2i  3027  dffo5  5616  halfnq  7331  nsmallnq  7333  0npr  7403  genpml  7437  genpmu  7438  ltexprlemm  7520  ltexprlemloc  7527  dedekindeulemlub  12998  dedekindicclemlub  13007
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