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Theorem r19.23 2659
Description: Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (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 2658 . 2  |-  ( F/ x ps  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph 
->  ps ) ) )
31, 2ax-mp 8 1  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1531   A.wral 2544   E.wrex 2545
This theorem is referenced by:  r19.23v  2660  rexlimi  2661  rexlimd  2665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-11 1716
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-ral 2549  df-rex 2550
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