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Theorem r19.21 2642
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 2641 . 2  |-  ( F/ x ph  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  (
ph  ->  A. x  e.  A  ps ) ) )
31, 2ax-mp 8 1  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( ph  ->  A. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1534   A.wral 2556
This theorem is referenced by:  r19.21v  2643  rmo3f  23194  rmo4fOLD  23195  r19.32  28048  rmoanim  28060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-nf 1535  df-ral 2561
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