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Theorem rexlimi 2471
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 2417 . 2  |-  A. x  e.  A  ( ph  ->  ps )
3 rexlimi.1 . . 3  |-  F/ x ps
43r19.23 2469 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph  ->  ps ) )
52, 4mpbi 143 1  |-  ( E. x  e.  A  ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1390    e. wcel 1434   A.wral 2349   E.wrex 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354  df-rex 2355
This theorem is referenced by:  rexlimiv  2472  r19.29af2  2497  triun  3896  reusv1  4216  reusv3  4218  onintrab2im  4270  fun11iun  5178
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