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Theorem rexlimi 2661
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 2609 . 2  |-  A. x  e.  A  ( ph  ->  ps )
3 rexlimi.1 . . 3  |-  F/ x ps
43r19.23 2659 . 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 1532    e. wcel 1685   A.wral 2544   E.wrex 2545
This theorem is referenced by:  rexlimiv  2662  triun  4127  reusv1  4533  reusv3  4541  tfinds  4649  fun11iun  5458  iunfo  8156  iundom2g  8157  fsumcom2  12231  dfon2lem7  23546  rexlimib  24357  bwt2  24991  finminlem  25630  stirlinglem13  27234  stirlinglem14  27235  reuan  27337
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-11 1716
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1530  df-nf 1533  df-ral 2549  df-rex 2550
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