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Theorem 19.23tOLD 1841
Description: Obsolete proof of 19.23t 1821 as of 1-Jan-2018. (Contributed by NM, 7-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
19.23tOLD  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) ) )

Proof of Theorem 19.23tOLD
StepHypRef Expression
1 exim 1585 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
2 19.9t 1796 . . . 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 1811 . . 3  |-  F/ x F/ x ps
6 nfe1 1750 . . . . 5  |-  F/ x E. x ph
76a1i 11 . . . 4  |-  ( F/ x ps  ->  F/ x E. x ph )
8 id 21 . . . 4  |-  ( F/ x ps  ->  F/ x ps )
97, 8nfimd 1830 . . 3  |-  ( F/ x ps  ->  F/ x ( E. x ph  ->  ps ) )
10 19.8a 1765 . . . . 5  |-  ( ph  ->  E. x ph )
1110a1i 11 . . . 4  |-  ( F/ x ps  ->  ( ph  ->  E. x ph )
)
1211imim1d 72 . . 3  |-  ( F/ x ps  ->  (
( E. x ph  ->  ps )  ->  ( ph  ->  ps ) ) )
135, 9, 12alrimdd 1787 . 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 4    <-> wb 178   A.wal 1550   E.wex 1551   F/wnf 1554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-11 1764
This theorem depends on definitions:  df-bi 179  df-ex 1552  df-nf 1555
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