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Theorem 19.21t 1997
Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 1998. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
Assertion
Ref Expression
19.21t  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )

Proof of Theorem 19.21t
StepHypRef Expression
1 nfr 1962 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
2 alim 1694 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
31, 2syl9 73 . 2  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
) )
4 19.9t 1980 . . . 4  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
54imbi1d 323 . . 3  |-  ( F/ x ph  ->  (
( E. x ph  ->  A. x ps )  <->  (
ph  ->  A. x ps )
) )
6 19.38 1723 . . 3  |-  ( ( E. x ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
75, 6syl6bir 237 . 2  |-  ( F/ x ph  ->  (
( ph  ->  A. x ps )  ->  A. x
( ph  ->  ps )
) )
83, 7impbid 195 1  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1453   E.wex 1674   F/wnf 1678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679
This theorem is referenced by:  19.21  1998  19.23t  2002  nfimd  2011  sbal1  2300  sbal2  2301  r19.21t  2797  r19.21tOLD  2798  ceqsalt  3082  sbciegft  3310  bj-ceqsalt0  31528  bj-ceqsalt1  31529  wl-sbhbt  31928  wl-2sb6d  31934  wl-sbalnae  31938  ax12indalem  32562  ax12inda2ALT  32563
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