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Theorem 19.9t 1780
Description: A closed version of 19.9 1784. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.)
Assertion
Ref Expression
19.9t  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )

Proof of Theorem 19.9t
StepHypRef Expression
1 df-nf 1545 . . 3  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
2 19.9ht 1779 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  ->  ph ) )
31, 2sylbi 187 . 2  |-  ( F/ x ph  ->  ( E. x ph  ->  ph )
)
4 19.8a 1747 . 2  |-  ( ph  ->  E. x ph )
53, 4impbid1 194 1  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1540   E.wex 1541   F/wnf 1544
This theorem is referenced by:  19.9h  1781  19.9d  1783  19.9OLD  1785  19.21t  1796  19.23t  1801  19.23tOLD  1821  vtoclegft  2931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746
This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545
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