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Theorem 19.35 1590
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, 5-Aug-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 19.26 1583 . . . 4  |-  ( A. x ( ph  /\  -.  ps )  <->  ( A. x ph  /\  A. x  -.  ps ) )
2 annim 414 . . . . 5  |-  ( (
ph  /\  -.  ps )  <->  -.  ( ph  ->  ps ) )
32albii 1556 . . . 4  |-  ( A. x ( ph  /\  -.  ps )  <->  A. x  -.  ( ph  ->  ps ) )
4 alnex 1533 . . . . 5  |-  ( A. x  -.  ps  <->  -.  E. x ps )
54anbi2i 675 . . . 4  |-  ( ( A. x ph  /\  A. x  -.  ps )  <->  ( A. x ph  /\  -.  E. x ps )
)
61, 3, 53bitr3i 266 . . 3  |-  ( A. x  -.  ( ph  ->  ps )  <->  ( A. x ph  /\  -.  E. x ps ) )
7 alnex 1533 . . 3  |-  ( A. x  -.  ( ph  ->  ps )  <->  -.  E. x
( ph  ->  ps )
)
8 annim 414 . . 3  |-  ( ( A. x ph  /\  -.  E. x ps )  <->  -.  ( A. x ph  ->  E. x ps )
)
96, 7, 83bitr3i 266 . 2  |-  ( -. 
E. x ( ph  ->  ps )  <->  -.  ( A. x ph  ->  E. x ps ) )
109con4bii 288 1  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  E. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531
This theorem is referenced by:  19.35i  1591  19.35ri  1592  19.25  1593  19.43  1595  speimfw  1635  19.39  1650  19.24  1651  19.36  1819  19.37  1821  sbequi  2012  grothprim  8472  sbequiNEW7  29553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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