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Theorem 19.21t 1926
Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 1927. (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 1891 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
2 alim 1647 . . 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 1909 . . . 4  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
54imbi1d 315 . . 3  |-  ( F/ x ph  ->  (
( E. x ph  ->  A. x ps )  <->  (
ph  ->  A. x ps )
) )
6 19.38 1677 . . 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 1397   E.wex 1627   F/wnf 1631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-12 1872
This theorem depends on definitions:  df-bi 185  df-ex 1628  df-nf 1632
This theorem is referenced by:  19.21  1927  nfimd  1939  sbal1  2222  sbal2  2223  r19.21t  2793  r19.21tOLD  2794  ceqsalt  3074  sbciegft  3300  wl-sbhbt  30207  wl-2sb6d  30213  wl-sbalnae  30217  bj-ceqsalt0  34835  bj-ceqsalt1  34836  ax12indalem  35127  ax12inda2ALT  35128
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