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Theorem r19.21 2631
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 2630 . 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 1532   A.wral 2545
This theorem is referenced by:  r19.21v  2632  r19.32  27325  rmoanim  27337
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-11 1716
This theorem depends on definitions:  df-bi 179  df-an 362  df-nf 1533  df-ral 2550
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