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Theorem r19.23 2904
Description: Restricted quantifier version of 19.23 1966. See r19.23v 2905 for a version requiring fewer axioms. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)
Hypothesis
Ref Expression
r19.23.1  |-  F/ x ps
Assertion
Ref Expression
r19.23  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph  ->  ps ) )

Proof of Theorem r19.23
StepHypRef Expression
1 r19.23.1 . 2  |-  F/ x ps
2 r19.23t 2903 . 2  |-  ( F/ x ps  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph 
->  ps ) ) )
31, 2ax-mp 5 1  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   F/wnf 1663   A.wral 2775   E.wrex 2776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-12 1905
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-ral 2780  df-rex 2781
This theorem is referenced by:  r19.23vOLD  2906  rexlimi  2907  rexlimdOLD  2910  ss2iundf  36111
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