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Theorem 19.9t 1793
Description: A closed version of 19.9 1797. (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 1554 . . 3  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
2 19.9ht 1792 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  ->  ph ) )
31, 2sylbi 188 . 2  |-  ( F/ x ph  ->  ( E. x ph  ->  ph )
)
4 19.8a 1762 . 2  |-  ( ph  ->  E. x ph )
53, 4impbid1 195 1  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550   F/wnf 1553
This theorem is referenced by:  19.9h  1794  19.9d  1796  19.9OLD  1798  19.21t  1813  19.23t  1818  19.23tOLD  1838  spimt  1955  sbft  2115  vtoclegft  3023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nf 1554
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