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Theorem 19.9h 1577
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 1575 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  ->  ph ) )
2 19.9h.1 . . 3  |-  ( ph  ->  A. x ph )
31, 2mpg 1383 . 2  |-  ( E. x ph  ->  ph )
4 19.8a 1525 . 2  |-  ( ph  ->  E. x ph )
53, 4impbii 124 1  |-  ( E. x ph  <->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1285   E.wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  19.9  1578  excomim  1596  exdistrfor  1725  sbcof2  1735  ax11ev  1753  19.9v  1796  exists1  2041
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