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Theorem 19.9h 1794
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.) (Proof shortened by Wolf Lammen, 5-Jan-2018.)
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 19.9h.1 . . 3  |-  ( ph  ->  A. x ph )
21nfi 1560 . 2  |-  F/ x ph
3 19.9t 1793 . 2  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
42, 3ax-mp 8 1  |-  ( E. x ph  <->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550   F/wnf 1553
This theorem is referenced by:  19.9  1797  19.23hOLD  1839  cbv3hv  1878  bnj1131  29059  bnj1397  29107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nf 1554
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