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Theorem 19.9v 1723
Description: Special case of Theorem 19.9 of [Margaris] p. 89 (see 19.9 1836). Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 1724. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 1734. (Revised by Wolf Lammen, 4-Dec-2017.)
Assertion
Ref Expression
19.9v  |-  ( E. x ph  <->  ph )
Distinct variable group:    ph, x

Proof of Theorem 19.9v
StepHypRef Expression
1 ax5e 1677 . 2  |-  ( E. x ph  ->  ph )
2 ax-5 1675 . . 3  |-  ( ph  ->  A. x ph )
3219.8w 1719 . 2  |-  ( ph  ->  E. x ph )
41, 3impbii 188 1  |-  ( E. x ph  <->  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   E.wex 1591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714
This theorem depends on definitions:  df-bi 185  df-ex 1592
This theorem is referenced by:  19.3v  1724  19.23v  1927  zfcndpow  8983  volfiniune  27828  prter2  30213  rfcnnnub  30944  relopabVD  32656  bnj937  32784  bnj594  32924  bnj907  32977  bnj1128  33000  bnj1145  33003
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