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Theorem 19.35 1588
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 1581 . . . 4  |-  ( A. x ( ph  /\  -.  ps )  <->  ( A. x ph  /\  A. x  -.  ps ) )
2 annim 416 . . . . 5  |-  ( (
ph  /\  -.  ps )  <->  -.  ( ph  ->  ps ) )
32albii 1554 . . . 4  |-  ( A. x ( ph  /\  -.  ps )  <->  A. x  -.  ( ph  ->  ps ) )
4 alnex 1531 . . . . 5  |-  ( A. x  -.  ps  <->  -.  E. x ps )
54anbi2i 677 . . . 4  |-  ( ( A. x ph  /\  A. x  -.  ps )  <->  ( A. x ph  /\  -.  E. x ps )
)
61, 3, 53bitr3i 268 . . 3  |-  ( A. x  -.  ( ph  ->  ps )  <->  ( A. x ph  /\  -.  E. x ps ) )
7 alnex 1531 . . 3  |-  ( A. x  -.  ( ph  ->  ps )  <->  -.  E. x
( ph  ->  ps )
)
8 annim 416 . . 3  |-  ( ( A. x ph  /\  -.  E. x ps )  <->  -.  ( A. x ph  ->  E. x ps )
)
96, 7, 83bitr3i 268 . 2  |-  ( -. 
E. x ( ph  ->  ps )  <->  -.  ( A. x ph  ->  E. x ps ) )
109con4bii 290 1  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  E. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1528   E.wex 1529
This theorem is referenced by:  19.35i  1589  19.35ri  1590  19.25  1591  19.43  1593  speimfw  1627  19.39  1673  19.24  1674  19.36  1808  19.37  1810  sbequi  1998  grothprim  8451
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1530
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