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Theorem 19.9h 1739
Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)
Hypothesis
Ref Expression
19.9h.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
19.9h  |-  ( E. x ph  <->  ph )

Proof of Theorem 19.9h
StepHypRef Expression
1 19.9ht 1738 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  ->  ph ) )
2 19.9h.1 . . 3  |-  ( ph  ->  A. x ph )
31, 2mpg 1538 . 2  |-  ( E. x ph  ->  ph )
4 19.8a 1730 . 2  |-  ( ph  ->  E. x ph )
53, 4impbii 180 1  |-  ( E. x ph  <->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   E.wex 1531
This theorem is referenced by:  19.23h  1740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-ex 1532
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