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Theorem 19.23t 1797
Description: Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.)
Assertion
Ref Expression
19.23t  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) ) )

Proof of Theorem 19.23t
StepHypRef Expression
1 exim 1563 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
2 19.9t 1783 . . . 4  |-  ( F/ x ps  ->  ( E. x ps  <->  ps )
)
32imbi2d 309 . . 3  |-  ( F/ x ps  ->  (
( E. x ph  ->  E. x ps )  <->  ( E. x ph  ->  ps ) ) )
41, 3syl5ib 212 . 2  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  ps )
) )
5 nfnf1 1758 . . 3  |-  F/ x F/ x ps
6 nfe1 1707 . . . . 5  |-  F/ x E. x ph
76a1i 12 . . . 4  |-  ( F/ x ps  ->  F/ x E. x ph )
8 id 21 . . . 4  |-  ( F/ x ps  ->  F/ x ps )
97, 8nfimd 1762 . . 3  |-  ( F/ x ps  ->  F/ x ( E. x ph  ->  ps ) )
10 19.8a 1719 . . . . 5  |-  ( ph  ->  E. x ph )
1110a1i 12 . . . 4  |-  ( F/ x ps  ->  ( ph  ->  E. x ph )
)
1211imim1d 71 . . 3  |-  ( F/ x ps  ->  (
( E. x ph  ->  ps )  ->  ( ph  ->  ps ) ) )
135, 9, 12alrimdd 1749 . 2  |-  ( F/ x ps  ->  (
( E. x ph  ->  ps )  ->  A. x
( ph  ->  ps )
) )
144, 13impbid 185 1  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1528   E.wex 1529   F/wnf 1532
This theorem is referenced by:  19.23  1798  sbft  1970  r19.23t  2658  ceqsalt  2811  vtoclgft  2835  sbciegft  3022
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-11 1716
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1530  df-nf 1533
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