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Theorem 19.9t 1782
Description: A closed version of 19.9 1783. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
Assertion
Ref Expression
19.9t  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )

Proof of Theorem 19.9t
StepHypRef Expression
1 df-ex 1529 . . 3  |-  ( E. x ph  <->  -.  A. x  -.  ph )
2 id 19 . . . . . 6  |-  ( F/ x ph  ->  F/ x ph )
32nfnd 1760 . . . . 5  |-  ( F/ x ph  ->  F/ x  -.  ph )
43nfrd 1743 . . . 4  |-  ( F/ x ph  ->  ( -.  ph  ->  A. x  -.  ph ) )
54con1d 116 . . 3  |-  ( F/ x ph  ->  ( -.  A. x  -.  ph  ->  ph ) )
61, 5syl5bi 208 . 2  |-  ( F/ x ph  ->  ( E. x ph  ->  ph )
)
7 19.8a 1718 . 2  |-  ( ph  ->  E. x ph )
86, 7impbid1 194 1  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528   F/wnf 1531
This theorem is referenced by:  19.9  1783  19.9d  1784  19.23t  1796  vtoclegft  2855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-ex 1529  df-nf 1532
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