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Theorem 19.35 1732
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 1700 . . 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 1695 . . . 4  |-  ( E. x  -.  ph  <->  -.  A. x ph )
5 pm2.21 111 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
65eximi 1702 . . . 4  |-  ( E. x  -.  ph  ->  E. x ( ph  ->  ps ) )
74, 6sylbir 216 . . 3  |-  ( -. 
A. x ph  ->  E. x ( ph  ->  ps ) )
8 exa1 1706 . . 3  |-  ( E. x ps  ->  E. x
( ph  ->  ps )
)
97, 8ja 164 . 2  |-  ( ( A. x ph  ->  E. x ps )  ->  E. x ( ph  ->  ps ) )
103, 9impbii 190 1  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  E. x ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187   A.wal 1435   E.wex 1659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678
This theorem depends on definitions:  df-bi 188  df-ex 1660
This theorem is referenced by:  19.35i  1733  19.35ri  1734  19.25  1735  19.43  1737  speimfwOLD  1783  19.39  1804  19.24  1805  19.36v  1809  19.37v  1815  19.36  2018  19.37  2020  spimt  2058  grothprim  9248  bj-nalnaleximiOLD  31042  bj-exaleximi  31044  bj-spimt2  31090  bj-spimtv  31099  bj-snsetex  31347
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