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Theorem 19.9v 1719
Description: Special case of Theorem 19.9 of [Margaris] p. 89 (see 19.9 1832). Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 1720. (Contributed by NM, 28-May-1995.) (Revised by NM, 1-Aug-2017.) Remove a dependency on ax-7 1730. (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 1673 . 2  |-  ( E. x ph  ->  ph )
2 ax-5 1671 . . 3  |-  ( ph  ->  A. x ph )
3219.8w 1715 . 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 1587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710
This theorem depends on definitions:  df-bi 185  df-ex 1588
This theorem is referenced by:  19.3v  1720  zfcndpow  8895  volfiniune  26791  prter2  29175  rfcnnnub  29907  relopabVD  31970  bnj937  32098  bnj594  32238  bnj907  32291  bnj1128  32314  bnj1145  32317
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