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Theorem 19.9 2058
Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1882 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 𝑥𝜑
Assertion
Ref Expression
19.9 (∃𝑥𝜑𝜑)

Proof of Theorem 19.9
StepHypRef Expression
1 19.9.1 . 2 𝑥𝜑
2 19.9t 2057 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2ax-mp 5 1 (∃𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wex 1694  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032
This theorem depends on definitions:  df-bi 195  df-ex 1695  df-nf 1700
This theorem is referenced by:  exlimd  2072  19.19  2082  19.36  2083  19.41  2088  19.44  2091  19.45  2092  19.9h  2104  exists1  2548  dfid3  4944  fsplit  7146  bnj1189  30137  bj-exexbiex  31684  bj-exalbial  31686  ax6e2ndeq  37599  e2ebind  37603  ax6e2ndeqVD  37970  e2ebindVD  37973  e2ebindALT  37990  ax6e2ndeqALT  37992
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