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Theorem 19.9 1947
Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1805 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)
Hypothesis
Ref Expression
19.9.1  |-  F/ x ph
Assertion
Ref Expression
19.9  |-  ( E. x ph  <->  ph )

Proof of Theorem 19.9
StepHypRef Expression
1 19.9.1 . 2  |-  F/ x ph
2 19.9t 1946 . 2  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
31, 2ax-mp 5 1  |-  ( E. x ph  <->  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   E.wex 1657   F/wnf 1661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909
This theorem depends on definitions:  df-bi 188  df-ex 1658  df-nf 1662
This theorem is referenced by:  19.9h  1948  exlimd  1974  19.19  2017  19.36  2021  19.44  2026  19.45  2027  19.41  2028  exists1  2367  dfid3  4712  fsplit  6856  bnj1189  29770  bj-exexbiex  31200  bj-exalbial  31202  ax6e2ndeq  36839  e2ebind  36843  ax6e2ndeqVD  37222  e2ebindVD  37225  e2ebindALT  37242  ax6e2ndeqALT  37244
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