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Theorem rexlimi 2631
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Hypotheses
Ref Expression
rexlimi.1  |-  F/ x ps
rexlimi.2  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
rexlimi  |-  ( E. x  e.  A  ph  ->  ps )

Proof of Theorem rexlimi
StepHypRef Expression
1 rexlimi.2 . . 3  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
21rgen 2579 . 2  |-  A. x  e.  A  ( ph  ->  ps )
3 rexlimi.1 . . 3  |-  F/ x ps
43r19.23 2629 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph  ->  ps ) )
52, 4mpbi 201 1  |-  ( E. x  e.  A  ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 6   F/wnf 1539    e. wcel 1621   A.wral 2516   E.wrex 2517
This theorem is referenced by:  rexlimiv  2632  triun  4066  reusv1  4471  reusv3  4479  tfinds  4587  fun11iun  5396  iunfo  8094  iundom2g  8095  fsumcom2  12167  dfon2lem7  23479  rexlimib  24290  bwt2  24924  finminlem  25563
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540  df-ral 2520  df-rex 2521
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