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Theorem r19.21t 2783
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 bi2.04 351 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ph  ->  ( x  e.  A  ->  ps ) ) )
21albii 1575 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  A. x
( ph  ->  ( x  e.  A  ->  ps ) ) )
3 19.21t 1813 . . 3  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ( x  e.  A  ->  ps ) )  <->  ( ph  ->  A. x ( x  e.  A  ->  ps ) ) ) )
42, 3syl5bb 249 . 2  |-  ( F/ x ph  ->  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  ( ph  ->  A. x ( x  e.  A  ->  ps ) ) ) )
5 df-ral 2702 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
6 df-ral 2702 . . 3  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
76imbi2i 304 . 2  |-  ( (
ph  ->  A. x  e.  A  ps )  <->  ( ph  ->  A. x ( x  e.  A  ->  ps )
) )
84, 5, 73bitr4g 280 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 4    <-> wb 177   A.wal 1549   F/wnf 1553    e. wcel 1725   A.wral 2697
This theorem is referenced by:  r19.21  2784  riotasv3dOLD  6590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nf 1554  df-ral 2702
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