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Theorem 19.23t 1808
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 1565 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
2 19.9t 1794 . . . 4  |-  ( F/ x ps  ->  ( E. x ps  <->  ps )
)
32imbi2d 307 . . 3  |-  ( F/ x ps  ->  (
( E. x ph  ->  E. x ps )  <->  ( E. x ph  ->  ps ) ) )
41, 3syl5ib 210 . 2  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  ps )
) )
5 nfnf1 1769 . . 3  |-  F/ x F/ x ps
6 nfe1 1718 . . . . 5  |-  F/ x E. x ph
76a1i 10 . . . 4  |-  ( F/ x ps  ->  F/ x E. x ph )
8 id 19 . . . 4  |-  ( F/ x ps  ->  F/ x ps )
97, 8nfimd 1773 . . 3  |-  ( F/ x ps  ->  F/ x ( E. x ph  ->  ps ) )
10 19.8a 1730 . . . . 5  |-  ( ph  ->  E. x ph )
1110a1i 10 . . . 4  |-  ( F/ x ps  ->  ( ph  ->  E. x ph )
)
1211imim1d 69 . . 3  |-  ( F/ x ps  ->  (
( E. x ph  ->  ps )  ->  ( ph  ->  ps ) ) )
135, 9, 12alrimdd 1760 . 2  |-  ( F/ x ps  ->  (
( E. x ph  ->  ps )  ->  A. x
( ph  ->  ps )
) )
144, 13impbid 183 1  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   E.wex 1531   F/wnf 1534
This theorem is referenced by:  19.23  1809  sbft  1978  r19.23t  2670  ceqsalt  2823  vtoclgft  2847  sbciegft  3034  sbftNEW7  29531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535
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