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Theorem 19.35 1672
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 39 . . . 4  |-  ( ph  ->  ( ( ph  ->  ps )  ->  ps )
)
21aleximi 1638 . . 3  |-  ( A. x ph  ->  ( E. x ( ph  ->  ps )  ->  E. x ps ) )
32com12 31 . 2  |-  ( E. x ( ph  ->  ps )  ->  ( A. x ph  ->  E. x ps ) )
4 exnal 1633 . . . 4  |-  ( E. x  -.  ph  <->  -.  A. x ph )
5 pm2.21 108 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
65eximi 1641 . . . 4  |-  ( E. x  -.  ph  ->  E. x ( ph  ->  ps ) )
74, 6sylbir 213 . . 3  |-  ( -. 
A. x ph  ->  E. x ( ph  ->  ps ) )
8 exa1 1646 . . 3  |-  ( E. x ps  ->  E. x
( ph  ->  ps )
)
97, 8ja 161 . 2  |-  ( ( A. x ph  ->  E. x ps )  ->  E. x ( ph  ->  ps ) )
103, 9impbii 188 1  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  E. x ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1379   E.wex 1597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616
This theorem depends on definitions:  df-bi 185  df-ex 1598
This theorem is referenced by:  19.35i  1674  19.35ri  1675  19.25  1676  19.43  1678  speimfwOLD  1721  19.39  1742  19.24  1743  19.36v  1747  19.37v  1753  19.36  1948  19.37  1950  spimt  1989  grothprim  9210  bj-nalnaleximiOLD  33936  bj-exaleximi  33938  bj-spimt2  33983  bj-spimtv  33992  bj-snsetex  34233
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