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Theorem 19.21t 1963
Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 1964. (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 1928 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
2 alim 1677 . . 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 1946 . . . 4  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
54imbi1d 318 . . 3  |-  ( F/ x ph  ->  (
( E. x ph  ->  A. x ps )  <->  (
ph  ->  A. x ps )
) )
6 19.38 1706 . . 3  |-  ( ( E. x ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
75, 6syl6bir 232 . 2  |-  ( F/ x ph  ->  (
( ph  ->  A. x ps )  ->  A. x
( ph  ->  ps )
) )
83, 7impbid 193 1  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435   E.wex 1657   F/wnf 1661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909
This theorem depends on definitions:  df-bi 188  df-ex 1658  df-nf 1662
This theorem is referenced by:  19.21  1964  19.23t  1968  nfimd  1977  sbal1  2259  sbal2  2260  r19.21t  2819  r19.21tOLD  2820  ceqsalt  3104  sbciegft  3330  bj-ceqsalt0  31452  bj-ceqsalt1  31453  wl-sbhbt  31846  wl-2sb6d  31852  wl-sbalnae  31856  ax12indalem  32485  ax12inda2ALT  32486
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