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Theorem 19.9 2070
Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1894 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 2069 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2ax-mp 5 1 (∃𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wex 1702  wnf 1706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-ex 1703  df-nf 1708
This theorem is referenced by:  exlimd  2085  19.19  2095  19.36  2096  19.41  2101  19.44  2104  19.45  2105  19.9h  2118  exists1  2559  dfid3  5015  fsplit  7267  bnj1189  31051  bj-exexbiex  32666  bj-exalbial  32668  ax6e2ndeq  38595  e2ebind  38599  ax6e2ndeqVD  38965  e2ebindVD  38968  e2ebindALT  38985  ax6e2ndeqALT  38987
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