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Theorem 19.35 1751
Description: Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
Assertion
Ref Expression
19.35  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  E. x ps )
)

Proof of Theorem 19.35
StepHypRef Expression
1 pm2.27 40 . . . 4  |-  ( ph  ->  ( ( ph  ->  ps )  ->  ps )
)
21aleximi 1715 . . 3  |-  ( A. x ph  ->  ( E. x ( ph  ->  ps )  ->  E. x ps ) )
32com12 32 . 2  |-  ( E. x ( ph  ->  ps )  ->  ( A. x ph  ->  E. x ps ) )
4 exnal 1710 . . . 4  |-  ( E. x  -.  ph  <->  -.  A. x ph )
5 pm2.21 112 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
65eximi 1718 . . . 4  |-  ( E. x  -.  ph  ->  E. x ( ph  ->  ps ) )
74, 6sylbir 218 . . 3  |-  ( -. 
A. x ph  ->  E. x ( ph  ->  ps ) )
8 exa1 1722 . . 3  |-  ( E. x ps  ->  E. x
( ph  ->  ps )
)
97, 8ja 166 . 2  |-  ( ( A. x ph  ->  E. x ps )  ->  E. x ( ph  ->  ps ) )
103, 9impbii 192 1  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  E. x ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189   A.wal 1453   E.wex 1674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693
This theorem depends on definitions:  df-bi 190  df-ex 1675
This theorem is referenced by:  19.35i  1752  19.35ri  1753  19.25  1754  19.43  1756  speimfwALT  1805  19.39  1826  19.24  1827  19.36v  1831  19.37v  1837  19.36  2055  19.37  2057  spimt  2108  grothprim  9290  bj-nalnaleximiOLD  31268  bj-spimt2  31356  bj-spimtv  31365  bj-snsetex  31603
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