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Theorem 19.9h 1534
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 1532 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  ->  ph ) )
2 19.9h.1 . . 3  |-  ( ph  ->  A. x ph )
31, 2mpg 1340 . 2  |-  ( E. x ph  ->  ph )
4 19.8a 1482 . 2  |-  ( ph  ->  E. x ph )
53, 4impbii 117 1  |-  ( E. x ph  <->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98   A.wal 1241   E.wex 1381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  19.9  1535  excomim  1553  exdistrfor  1681  sbcof2  1691  ax11ev  1709  19.9v  1751  exists1  1996
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