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Theorem 19.9d 1941
Description: A deduction version of one direction of 19.9 1943. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)
Hypothesis
Ref Expression
19.9d.1  |-  ( ps 
->  F/ x ph )
Assertion
Ref Expression
19.9d  |-  ( ps 
->  ( E. x ph  ->  ph ) )

Proof of Theorem 19.9d
StepHypRef Expression
1 19.9d.1 . . 3  |-  ( ps 
->  F/ x ph )
2 df-nf 1664 . . 3  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
31, 2sylib 199 . 2  |-  ( ps 
->  A. x ( ph  ->  A. x ph )
)
4 19.9ht 1940 . 2  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  ->  ph ) )
53, 4syl 17 1  |-  ( ps 
->  ( E. x ph  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435   E.wex 1659   F/wnf 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-12 1905
This theorem depends on definitions:  df-bi 188  df-ex 1660  df-nf 1664
This theorem is referenced by:  19.9t  1942  exdistrf  2130  equveli  2143  copsexg  4703  19.9d2rf  28098  wl-exeq  31781
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