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Theorem 19.9ht 1726
Description: A closed version of 19.9 1783. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
19.9ht  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  ->  ph ) )

Proof of Theorem 19.9ht
StepHypRef Expression
1 df-ex 1529 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
2 hbnt 1724 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)
32con1d 116 . 2  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  A. x  -.  ph  ->  ph ) )
41, 3syl5bi 208 1  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  ->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527   E.wex 1528
This theorem is referenced by:  19.9h  1727
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
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