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Theorem 19.21t 1843
Description: Closed form of Theorem 19.21 of [Margaris] p. 90. (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 1812 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
2 alim 1604 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
31, 2syl9 71 . 2  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
) )
4 19.9t 1829 . . . 4  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
54imbi1d 317 . . 3  |-  ( F/ x ph  ->  (
( E. x ph  ->  A. x ps )  <->  (
ph  ->  A. x ps )
) )
6 19.38 1630 . . 3  |-  ( ( E. x ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
75, 6syl6bir 229 . 2  |-  ( F/ x ph  ->  (
( ph  ->  A. x ps )  ->  A. x
( ph  ->  ps )
) )
83, 7impbid 191 1  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368   E.wex 1587   F/wnf 1590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794
This theorem depends on definitions:  df-bi 185  df-ex 1588  df-nf 1591
This theorem is referenced by:  19.21  1844  nfimd  1855  sbal1  2180  sbal2  2182  sbal2OLD  2193  ax12indalem  2255  ax12inda2ALT  2256  r19.21t  2907  ceqsalt  3101  sbciegft  3325  wl-sbhbt  28527  wl-2sb6d  28533  wl-sbalnae  28537  bj-ceqsalt0  32717  bj-ceqsalt1  32718
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