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Theorem r19.23 2629
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 2628 . 2  |-  ( F/ x ps  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph 
->  ps ) ) )
31, 2ax-mp 10 1  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   F/wnf 1539   A.wral 2516   E.wrex 2517
This theorem is referenced by:  r19.23v  2630  rexlimi  2631  rexlimd  2635
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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