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Theorem r19.21 2591
Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)
Hypothesis
Ref Expression
r19.21.1  |-  F/ x ph
Assertion
Ref Expression
r19.21  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( ph  ->  A. x  e.  A  ps ) )

Proof of Theorem r19.21
StepHypRef Expression
1 r19.21.1 . 2  |-  F/ x ph
2 r19.21t 2590 . 2  |-  ( F/ x ph  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  (
ph  ->  A. x  e.  A  ps ) ) )
31, 2ax-mp 10 1  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( ph  ->  A. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   F/wnf 1539   A.wral 2509
This theorem is referenced by:  r19.21v  2592
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-nf 1540  df-ral 2513
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