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Theorem r19.21t 2629
Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)
Assertion
Ref Expression
r19.21t  |-  ( F/ x ph  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  (
ph  ->  A. x  e.  A  ps ) ) )

Proof of Theorem r19.21t
StepHypRef Expression
1 bi2.04 352 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ph  ->  ( x  e.  A  ->  ps ) ) )
21albii 1558 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  A. x
( ph  ->  ( x  e.  A  ->  ps ) ) )
3 19.21t 1794 . . 3  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ( x  e.  A  ->  ps ) )  <->  ( ph  ->  A. x ( x  e.  A  ->  ps ) ) ) )
42, 3syl5bb 250 . 2  |-  ( F/ x ph  ->  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  ( ph  ->  A. x ( x  e.  A  ->  ps ) ) ) )
5 df-ral 2549 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
6 df-ral 2549 . . 3  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
76imbi2i 305 . 2  |-  ( (
ph  ->  A. x  e.  A  ps )  <->  ( ph  ->  A. x ( x  e.  A  ->  ps )
) )
84, 5, 73bitr4g 281 1  |-  ( F/ x ph  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  (
ph  ->  A. x  e.  A  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532   F/wnf 1536    e. wcel 1688   A.wral 2544
This theorem is referenced by:  r19.21  2630  riotasv3dOLD  6349
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-11 1719
This theorem depends on definitions:  df-bi 179  df-an 362  df-nf 1537  df-ral 2549
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